Algebraic geometry codes from higher dimensional varieties

ALGEBRAIC GEOMETR Y CODES FR OM HIGHER DIMENSIONAL V ARIETIES JOHN B. LITTLE Abstract. This paper is a general surv ey of work on Gopp a-t ype co des from higher dimensional algebraic v arieties. The con struction and sev eral tec hniques for estimating the mini mum di stance are describ ed ﬁrst. Codes from v arious classes of v ari eties, including Hermitian h yp ersurfaces, Grass m annians, ﬂag v arieties, ruled surfaces o ver curves, and Deligne-Lusztig v arieties are consid- ered. Connec tions with the theories of toric codes and order domains are also brieﬂy indicated. 1. Introduction The co des considered in this survey ca n all b e understo o d as examples of eval- uation c o des pro duced from a ﬁnite set S = { P 1 , . . . , P n } of F q -rational p o in ts o n an alg ebraic v ar iet y X and an F q -vector space of functions F deﬁned on S . The set of c o dewords is the imag e of an ev alua tion mapping ev S : F − → F n q (1) f 7→ ( f ( P 1 ) , . . . , f ( P n )) . X will usually b e assumed smo oth, but in fact many of the constructions also make sense for normal v arie ties (m uch of the usual g eometric theory of divisors and line bundles on normal v ar ieties is the same as in the smo oth case). The Goppa C L ( D , G ) codes from curves X where F = L ( G ) for some divisor G on X were the ﬁr s t exa mples of co des of this type to b e considered. Rela tiv ely early in the histo ry of a pplications of alg ebraic g eometry to co ding theory , how e v er , Tsfasman and Vladut pro p osed in Chapter 3.1 of [54] that higher dimensional v a rieties migh t a lso be used to construct co des. By the results of [45], every linear co de can b e o btained by the cons truction of Deﬁnition 1 b elow, starting from some S ⊆ X ( F q ) for some v arie ty X a nd s ome line bundle L o n X ; indee d curves suﬃce for this (see Sectio n 9). Hence the ques tion is whether one can identify sp eciﬁc higher dimensional v arieties X , spaces o f functions F , and sets of r ational p oints S that yield particularly int e resting co de s using alg ebraic geometric c o nstructions. There ha s b een a fairly steady stream of articles since the 1 990’s studying s uc h co des and our ﬁrst main goal here is to survey the methods tha t ha ve b een develop ed and the res ults that hav e b een obtained. In a sense, the ﬁrst ma jor diﬀerence b et ween higher dimensional v a rieties and curves is that p oints on X o f dimension ≥ 2 are sub v ar ieties of co dimension ≥ 2, not divisor s. This mea ns that ma n y of the fa miliar to ols used for Goppa co des 2000 Mathematics Subject Classiﬁc ation. Primary 94B27; Seconda ry 14G50, 14J99. Key wor ds and phr ases. coding theory , Goppa co de, quadric, Hermi tian v ariet y , Grass m annian, ﬂag v ariety , Del P ezzo s urface, ruled surface, Deligne-Lusztig v ariety . 1 2 JOHN B. LITTLE (e.g. Riemann-Ro ch theorems, the theory of diﬀeren tials and residues, etc.) do no t apply in e xactly the sa me w ay . A second diﬀerence is the poss ibility o f per forming bir ational mo diﬁc ations such as blowing up p oin ts or other s ub v arieties on a v ariety of higher dimension. F or instance, if p is a point in a smo oth algebraic v arie t y X o f dimension δ ≥ 2, there is another smo oth v a riet y Y = Bl p ( X ), a prop er morphism π : Y → X , and an exc eptional divisor E ≃ P δ − 1 in Y such that π ( E ) = { p } , a nd π | Y − E : Y − E ≃ X − { p } as v ar ieties. Beca use Y and X hav e isomorphic nonempty Zariski-op en subsets, they have isomo r phic function ﬁelds. Such v ar ieties Y and X are said to be bir ational ly isomorphic . This says that function ﬁelds in tw o or more v ar iables alwa ys ha ve many diﬀerent nonisomorphic smo oth models, and the connection with function ﬁelds is not as tight as in the curve case. It must be said that the theor y of Goppa - t yp e co des from higher dimens io nal v a rieties is muc h less adv a nced at this p oint than the theory for Go ppa co des from curves, p erhaps b ecause of these diﬀerences . The r e is still no clear understanding of how b est to harness the prop erties of higher dimensiona l v a rieties in co ding theory . Indeed, as we w ill see, most of the work that has app eared to date has b een devoted to cas e studies of the st ructur al pr op erties o f codes co nstructed from certain particular families of v ar ie ties X – their par ameters, their weight distr ibutions, their hierar c hies of higher Hamming weights, a nd so forth. A few general ideas for estimating the minimum dista nc e d ha ve b een developed. Ho wev er , quite a few of the co des that we will see are r ather unremar k able; in many of the ca ses w her e the exact weigh t distributions are known, other algebraic constructions yield b etter co des. In addition, the development of eﬃcient enco ding a nd deco ding a lg orithms for these co des has not really b egun (see Sectio n 9 on this p oint, though). The theory of order domains should yield to ols here as well as for co des from curves. Nevertheless, the universalit y of this constructio n oﬀers hop e that go o d examples can be co nstructed this way , and o ur sec o nd main goa l is to encoura ge others to explore this a rea. This survey is orga nized as follows. In Section 2, we give tw o v ar ian ts of Tsfasman and Vladut’s co de cons truction, one star ting from an a bstract v ar ie ty X and line bundle L on X , the other star ting fro m a n embedded v ariety X ⊂ P m . W e also present some ﬁrst examples. F our general metho ds for estimating the minimum distance a re presented in Section 3. Two app eared ﬁr s t in S.H. Hanse n’s ar ticle [26]. F or the ﬁrst of these, it is assumed that all of the F q -rational po ints of interest are cont ained in a family of curv es on X and intersection pro ducts of divisors with those cur v es are used to b ound d . The second metho d is based on the Sesha dri constant of the line bundle L with r espect to the set of F q -rational points on X . A third metho d fro m [17] can b e used when the set o f F q -rational p oin ts is itself a complete in tersectio n in P m . Finally , we present another, mor e arithmetic, metho d based on the W eil conjectures develop ed b y Lachaud in [37]. The ne x t sections 4 a nd 5 present a selection of the exa mples of these co des that hav e a ppear ed in the literature, co des constr ucted from quadric hypersurfa c es, Hermitian hypersurfa c es, Grassma nnians and ﬂag v arieties, Del Pezzo sur faces, ruled surfaces, and Deligne-Lusztig v a rieties. Finally , we presen t so me comparisons betw een co des in Sectio n 7. Where practicable, we ha ve provided brief pro ofs of the results w e state, in order to show the metho ds inv o lv ed in the study of thes e co des. CODES FROM HIGHER DIMENSIONAL V ARIETIES 3 As we pro ceed through these examples, the pre r equisites from algebraic geometry steadily increas e. Our in tended audience includes b oth co ding theoris ts familiar with the theory of Goppa co des o n cur v es but not higher dimensiona l geo metry and algebra ic g eometers curious ab out how higher dimensiona l v arie ties migh t be used in the co ding theory context. Hence there are proba bly p ortions of what we say that might seem unnecessar ily elemen ta ry to some r eaders. W e ap olog iz e in adv ance. The text [2 8] by Ha rtshorne is a go o d g eneral re ference for most of the alge- braic geometry w e need. The construction of Grassmannians v ia ex terior a lgebra, Sch ub e rt v arie ties , and the intersection theory on Gra ssmannians a r e covered in Griﬃths and Harr is, [19]. A full understanding of the Deligne-Lusztig v arieties also depe nds on the theo ry of reductive algebr a ic groups G over ﬁelds o f characteristic p and the classiﬁcation o f their ﬁnite subgroups G F by ro ot systems and Dynkin diagrams with an action o f the F rob enius endomo rphism, F . The b oo k [5] of Carter contains a ll the informa tion ne e ded for this. Because of spa ce limitations, it has no t b een p ossible to discuss all the results of every pa per in this area in detail. Pointers to a ll of the litera ture o f which the author is aw ar e are provided in the bibliogra phic notes in Section 9, the references, and their bibliographies. An y omissions or errors are entirely due to the author. Any co mmen ts or sug- gestions are w elco me. 1.1. Notation. W e will use the following gener al no tational and termino lo gical conv entions. • The num b er of element s in a ﬁnite set T will b e denoted b y # T . • The p ar ameters o f a linear co de ar e denoted [ n, k , d ] as usua l, w he r e n is the blo ck length, k is the dimension, a nd d is the minim um distance. • The gener alize d H amming weights are deno ted d r , 1 ≤ r ≤ k . As in [57], d r is the s ize of the minimal s upp ort of an r - dimensional sub co de of C , extending the usual minimum distance d = d 1 . • W e denote an algebraica lly closed ﬁeld of c har acteristic p by F a nd all ﬁnite ﬁelds F q for q = p m are cons idered as subﬁelds o f F . • The pr oje ctive sp ac es P m , Gra ssm annians G ( ℓ, m ), and so for th are con- sidered as v arieties o ver the algebraically clo sed ﬁeld F in order to “do geometry .” The F q -rational po in ts used in the constr uction of the co des are ﬁnite subsets of these v a rieties. • If f is a homogeneous polyno mial in F q [ x 0 , . . . , x m ], V ( f ) is the ze ro locus of f in P m . • A line bund le is a lo cally free sheaf of rank one. A t sev eral points, it will be conv enient to use the she af c ohomolo gy groups H i ( X, L ) for a line bundle L . The space o f global sections will also b e written Γ( X, L ). 2. The General Construction Several appar en tly diﬀeren t, but essentially equiv alent, v er sions of the construc- tion are commonly encountered in the literature. F or instance, o ne de s cription starts from a smo oth pro jectiv e v a riet y X deﬁned over F q , a set S ⊆ X ( F q ) o f F q -rational p oints of X , and a line bundle L on X , also deﬁned o ver F q . Le t P be 4 JOHN B. LITTLE an F q -rational p oint of X . The stalk L P , mo dulo sections v anishing at P , denoted L P , is is o morphic to F q by a choice of lo cal trivialization. Deﬁnition 1. The choice o f such loc a l trivializations at each point in S deﬁnes a linear mapping (called the germ map in [54]) (2) α : Γ( X, L ) − → n M i =1 L P i ≃ F n q , and the image is the co de deno ted C ( X , L ; S ), or C ( X , L ) if the set of p oints S is understo o d from the context. If L = O X ( D ) for an F q -rational divisor D on X whos e s upport is disjoin t fr om { P 1 , . . . , P n } , then up to monomial equiv alence, this is the s a me as the ev alua tion co de as in (1 ) from the subspace F of the ﬁeld of r ational functions of X given b y F = { f ∈ F q ( X ) ∗ : div ( f ) + D ≥ 0 } ∪ { 0 } . F or instance, when X is a smooth algebraic curve and L = O X ( G ) fo r some diviso r G deﬁned ov er F q whose supp ort is disjoint from the supp ort of D = P 1 + · · · + P n , then this is the same a s the algebr aic geometric Goppa co de C L ( D , G ) from X . F or explicit constructio ns of co des from embedded v arieties X ⊆ P m , a nother more elementary desc r iption is also av ailable using homo gene ous c o or dinates ( a 0 : a 1 : · · · : a m ) fo r po in ts in P m , where ( a 0 : a 1 : · · · : a m ) and ( λa 0 : λa 1 : · · · : λa m ) represent the same p oint whenev e r λ ∈ F ∗ . Deﬁnition 2. Cho osing any o ne such homogeneous co ordinate vector deﬁned ov er F q for each of the p oints P i in the set S , deﬁne an ev a luation map ev S and a code as in (1) us ing the vector space F 1 of linear forms (homogeneous p olynomials o f degree 1) in F q [ x 0 , . . . , x m ]. The code obtained as the image of this mapping is often denoted C ( X ), o r C ( X ; S ) if it is impor tan t to sp ecify the s e t of p oints. Similarly , the space of linear forms can be replaced by the vector spa ce F h of homogeneo us po lynomials of any degree h ≥ 1, and corr esponding co des denoted C h ( X ; S ) or C h ( X ) are obtained. Example 1. Let X = P m itself, and let S b e the set of aﬃne F q -rational p oints of X , that is, po ints in the complement of the h yp erplane V ( x 0 ), ha ving ho mogeneous co ordinate vectors of the form (1 : a 1 : . . . : a m ). With these particular co ordina te vectors, the co de C h ( X ; S ) is the well-kno wn q -ary h th or der (g e ne r alized) R e e d- Mul ler co de, denoted R q ( h, m ). (When m = 1 , this is the same as an ex tende d R e e d-Solomon co de.) The blo c k length is n = q m . If h < q , then the monomials x β = x β 0 0 · · · x β m m where | β | = β 0 + · · · + β m = h are linear ly indep endent o n S , so the dimensio n of R q ( s, m ) is k =  m + h h  . If S = P m ( F q ), the resulting pr oje ctive R e e d-Mul ler c o des hav e blo ck length n = q m + · · · + q + 1 . ♦ There is, of co urse, a tight connectio n b et ween Deﬁnition 1 and Deﬁnition 2. If X is embedded in P m and L = O X (1) is the hyperplane section bundle, then C ( X, O X (1)) and C ( X ) a re mo nomially equiv alent co des (they diﬀer at most by constant multiples in each component depending on how the isomorphisms of the ﬁber s with F q are chosen). Similarly , C h ( X ) is equiv alent to C ( X , O X ( h )). Also, in theory it s uﬃces to consider the C ( X ) = C 1 ( X ) co des, since the C h ( X ) c ode on X is the same as the C 1 co de o n the v ariety ν h ( X ), where ν h is the degree- h CODES FROM HIGHER DIMENSIONAL V ARIETIES 5 V er onese mapping ν h : P m − → P ( m + h h ) − 1 ( x 0 : x 1 : · · · : x m ) 7→ ( · · · : x β : · · · ) , and x β = x β 0 0 · · · x β m m ranges o ver all monomials of tota l degree h . The imag e ν h ( P m ) has dimens ion m , degr ee h m , and is isomorphic to P m . 3. Estima ting the P arameters 3.1. Elementary b ounds . Suppos e Deﬁnition 2 is used to construct a co de C h ( X ; S ) from a v ariety X . The block length o f the co de is n = # S . Using a standar d linear algebra result, the dimension is k = dim F h − dim k e r ev S . F orms of degree h v anishing on X a lw ays give element s of the kernel. The dimen- sion o f the space of such forms can b e computed using the long exa ct cohomolo gy sequence of (3) 0 − → I X ( h ) − → O P m ( h ) − → O X ( h ) − → 0 . Since each co dew ord is ev S ( f ) = ( f ( P 1 ) , . . . , f ( P n )) for so me for m f , the co de - word weigh t is n − #( V ( f ) ∩ S ), the n umber of P i in S where f is not zero. Therefore, (4) d = min f 6 =0 ∈F h ( n − #( V ( f ) ∩ S )) . Along similarly general lines, let dim Y = δ a nd let the degree of Y be s < q + 1 in P m . Let E b e an F q -rational linear subspace of dimensio n m − δ − 1 w ith E ∩ Y = ∅ . By pro jection from E onto a linear s ubs pa ce L ≃ P δ , eac h F q -rational point of L corres p onds to at mo s t s such p oin ts of Y , so (5) # Y ( F q ) ≤ s · # P δ ( F q ) = s ( q δ + · · · + q + 1) . Applying (5) to Y = X ∩ H for a hyperplane, L ac ha ud obtains the following elementary b ound in [37]. Theorem 1. L et X b e a pr oje ctive variety of dimension δ and de gr e e s < q + 1 . Then for h = 1 the C ( X ) c o de has d ≥ n − s ( q δ − 1 + · · · + q + 1) . A mor e r eﬁned e stimate of the num b er of F q rational p oin ts on a pro jective hypersurface e stablishes the following re s ult for the pro jectiv e Reed-Muller co des int r oduced in Example 1. Theorem 2. L et h ≤ q . The pr oje ct ive R e e d-Mul ler c o de of or der h has p ar ameters  q m + · · · + q + 1 ,  m + h h  , ( q + 1 − h ) q m − 1  . Pr o of. W rite S = P m ( F q ). The ev a luation mapping is injective and k = dim F s =  m + h h  provided that d > 0. By [5 1], if f is a homogeneous polyno mia l of degree h ≤ q , then (improving the b ound of (5)) #( V ( f ) ∩ S ) ≤ hq m − 1 + q m − 2 + · · · + q + 1 . 6 JOHN B. LITTLE Moreov er, if V ( f ) is the union of h F q -rational hyperplanes meeting along a common ( m − 2)-dimensional linea r subspace, this b ound is attained. Hence d = ( q m + q m − 1 + · · · + q + 1) − ( hq m − 1 + q m − 2 + · · · + q + 1 ) = ( q + 1 − h ) q m − 1 as claimed.  In the r e mainder of this s ection, several other gener a l techniques for estima t- ing the minimum dista nc e of these co des will be co nsidered. The ﬁrst three are primarily geometr ic, while the last is arithmetic in nature. 3.2. Bounds from cov ering families of curves. F or the following discus sion, it will b e most conv enient to use the co de construction given in Deﬁnition 1. In many concr ete cases, it can be seen that the points in the set S are distr ibuted on a collectio n of curves C i (subv arieties o f dimension 1) on the v arie t y X . Since ea c h section f ∈ Γ( X , L ) on X deﬁnes a divis or of zero es Z ( f ), a subv ar iet y o f co di- mension 1 on X , determining the minim um distance of the C ( X , L ) co de reduces to understanding ho w many times the divisors Z ( f ) can intersect the cur v es C i at po in ts o f S . T o prepar e, let C be any ir reducible curve in X . O bserve that the divisors Z ( f ) for f ∈ Γ( X , L ) all cut out divisors on C o f the same deg ree. T his degree will b e denoted by L · C . In this situation, Hansen der iv es a lower bound for d in [26]. Theorem 3. L et X b e a normal pr oje ct ive variety deﬁne d over F q , of dimension dim X ≥ 2 . L et S ⊆ X ( F q ) and assume S ⊂ a [ i =1 C i wher e C i ar e irr e ducible curves on X , also deﬁne d over F q . Assume t hat #( C i ∩ S ) ≤ N for al l i . L et L b e a line bund le on X deﬁne d over F q such that 0 ≤ L · C i ≤ η ≤ N for al l i . L et ℓ = max f 6 =0 ∈ Γ( X, L ) # { i : Z ( f ) c ont ains C i } . Then the c o de C ( X , L ; S ) has d ≥ # S − ℓN − ( a − ℓ ) η . Pr o of. Let f ∈ Γ( X , L ), let D = Z ( f ), a nd let E = Z ( f ) ∩ a [ i =1 C i . Supp ose E contains ℓ ′ ≤ ℓ o f the C i . The num b er p oin ts of S that are contained in E is estimated as follows: #( E ∩ S ) ≤ ℓ ′ N + ( a − ℓ ′ ) η ≤ ℓN + ( a − ℓ ) η (since b y h yp o thesis η ≤ N ). Hence ev S ( f ) has at least # S − ℓN − ( a − ℓ ) η nonzero ent ries.  Example 2. Let X = P 1 × P 1 . Let S = X ( F q ), w hich cons is ts of ( q + 1) 2 po in ts, equally distributed over the lines C 1 , . . . , C q +1 of one of the rulings. The Picar d group of line bundles mo dulo iso morphism is Pic( X ) ≃ Z ⊕ Z , s o the lines C i may be taken as the divisors of zeros of sections o f a line bundle of type (1 , 0 ). Let L hav e type ( α, β ) where 0 ≤ α, β ≤ q + 1 . Apply Theorem 3 to estimate d for the C ( X, L ) co de. Because of the descr iption o f S ab ov e, N = q + 1. The divisor Z ( f ) CODES FROM HIGHER DIMENSIONAL V ARIETIES 7 for f ∈ Γ( X , L ) cont ains at most α of the C i , so ℓ = α . Mo reov er, L · C i = β for each i , so η = β . The bound is d ≥ ( q + 1) 2 − α ( q + 1) − ( q + 1 − α ) β = ( q + 1 − α )( q + 1 − β ) . It is e asy to cons tr uct co dewords of this weigh t via bihomogeneous p olynomials on P 1 × P 1 . So this is the exact minimum distance. ♦ 3.3. Bounds using Se shadri constants. A second general method for es timating the minim um distance of the C ( X , L ; S ) co des is based on the Seshadri c onstant of L relative to the set S . This is potentially useful but requires some signiﬁcantly more sophisticated bira tional geo metry to state and a pply . Let π : Y → X b e the blow up of the X at the p oin ts in S a nd call the exceptiona l diviso r E . Then the Seshadri constant is deﬁned as ε ( L , S ) = sup { ε ∈ Q : π ∗ L − εE is nef on Y } . (Here, “nef ” means numeric al ly eﬀe ctive , that is, ( π ∗ L − εE ) · C ≥ 0 for all irre- ducible curves C on Y .) Hansen proves the following e s timate for the minimum distance of the C ( X , L ; S ) code s in [26]. Theorem 4. L et X b e a nonsingular pr oje ctive variety of dimension ≥ 2 over F q . If L is ample with Seshadri c onst ant ε ( L , S ) ≥ e ∈ N , and n > e 1 − dim( X ) L dim( X ) , then C ( X , L ; S ) has minimu m distanc e d ≥ n − e 1 − dim( X ) L dim( X ) . This is particula rly well-suited for analyzing cer tain c odes from Deligne-Lusztig v a rieties to b e deﬁned in Section 5 below. 3.4. Bounds from S itself. All of the C h ( X ; S ) co des introduced in Section 2 can b e viewed as punctures o f the pro jectiv e Reed-Muller co de of order h on the appropria te P m (delete the comp o nen ts corresp onding to p o in ts in the complemen t of S ). F or this r eason, in a ddition to ma k ing use of the prop erties of the v ariety X , it is als o pos sible to use prop erties of the 0- dimens io nal algebraic set (or scheme) S itself to estimate d . Let I P be the sheaf o f ideals deﬁning any 0-dimensio nal P . F rom the lo ng exact cohomo logy s equence of the exact sequence o f shea ves 0 − → I P − → O P m − → O P − → 0 , it follows that for a ll h ≥ 0, (6) 0 → H 0 ( I P ( h )) → H 0 ( O P m ( h )) → H 0 ( O P ( h )) → H 1 ( I P ( h )) → 0 . The term H 0 ( I P ( h )) gives the space of homog eneous for ms of degree h v anishing on P . The term H 1 ( I P ( h )) measures the failure of the po ints in P to impos e independent conditions on forms of degree h . In the case that S is a c omplete interse ct ion of hypers ur faces of degr ees d 1 , . . . , d m , there a re particularly nice techniques fro m commutativ e algebra a nd alg e braic ge- ometry rela ted to the cla s sical Cayle y-Bachar ach The or em that apply . A mo dern version o f this result due to Da vis, Geramita, and Orec c hia can b e stated as follows in the situatio n at hand. Theorem 5. L et S ⊂ P m b e a r e duc e d c omplete int erse ction of hyp ersurfac es of de gr e es d 1 , . . . , d m . L et Γ ′ , Γ ′′ b e disjoint su bsets of S with S = Γ ′ ∪ Γ ′′ . L et s = P m i =1 d i − m − 1 . Th en for al l h ≥ 0 , dim H 0 ( I Γ ′ ( h )) − dim H 0 ( I S ( h )) = dim H 1 ( I Γ ′′ ( s − h )) . 8 JOHN B. LITTLE Hence, one w ay to in terpr et Theor em 5 is that when Γ ′ ⊂ S , the diﬀerence in dimension betw een the space of homogeneous forms of deg ree a v anishing on Γ ′ and the subspace v anishing on S is equal to the dimension of H 1 ( I Γ ′′ ( s − a )). Moreover by (6), this dimension mea sures the failure o f Γ ′′ to imp ose independent conditions on homogene o us forms of degree s − a . Applied to the cor resp o nding co des from S consisting of d 1 d 2 . . . d m distinct F q -rational p oints, this result implies the following. Theorem 6. L et S b e a r e duc e d c omplete interse ction of hyp ersurfac es of de gr e es d 1 , . . . , d m in P m . L et s = P m i =1 d i − m − 1 as in The or em 5. If 1 ≤ h ≤ s , t he c o de C h ( S ) has minimum distanc e d ≥ m X i =1 d i − h − ( m − 1) = s − h + 2 . The pro of is ac complished b y showing that under these hypotheses , a n y form of degree h tha t is zer o on a subset Γ ′ that is too lar ge mu st b e zero a t all p oint s in S b ecause the H 1 ( I Γ ′′ ( s − h )) g roup v anishes. The b ound on d g iv en her e w as improv ed rather str ikingly b y Ballico and F onta- nari to d ≥ m ( s − h ) + 2 under the a ssumption that all subsets of m + 1 of the po in ts in S span P m – see [2 ] for this. Bounds derived by these methods are usually in ter esting only for h close to s . Mo reov er some, but not all, interesting examples of S s atisfy the co mplete int e rsection hypothesis. F or instance the a ﬃne F q -rational p oints in P m form a complete in ter section for all m . The F 8 -rational po in ts o n the K lein qua r tic and the F r 2 po in ts on the Hermitian curve a re other examples . 3.5. General W eil-t yp e b ounds. F ro m (4) above, and the pr oof of Theorem 2, the minimum distance of a C ( X ) co de as in Deﬁnition 2 is determined b y the nu m ber s o f F q -rational p o in ts o n the sub v ar ie ties Y = X ∩ V ( f ). Hence, a no ther po ssible approach to estimate d is to apply gener al b ounds for # Y ( F q ), for instance bo unds der iv ed fr o m the statements of the W eil conjecture s , or reﬁned v er sions of these. W e very brieﬂy recall the deep ma thematics b ehind this appro ac h. Thinking of X as a v a r iet y o ver the algebr aic closure of the ﬁnite ﬁeld, the n umber of F q -rational po in ts on X can be computed b y an analog of the L e fschetz trace formula for the action of the F r o benius endo morphism F o n the ℓ -adic ´ etale cohomolog y groups of X , H i ( X ) (where ℓ is an y prime not dividing q ): (7) # X ( F q ) = 2 m X i =0 ( − 1) i T r( F | H i ( X )) . Moreov er, the eigenv alues of F on H i ( X ) are algebraic n umber s o f abso lute v alue q i/ 2 . When X is obtained fro m a v ariety Y deﬁned ov e r the ring of integers R of some num b er ﬁeld b y reduction mo dulo some prime idea l in R , then the dimensions of the H i ( X ) ar e the same as the topolo gical Betti num b ers of the v ar iet y ov e r C corres p onding to Y . CODES FROM HIGHER DIMENSIONAL V ARIETIES 9 Thu s, for instance, if X is a s mo oth curve of genus g whic h is the reductio n o f a smo oth curve Y , then # X ( F q ) = 1 + q − 2 g X j =0 α j , where | α j | = q 1 / 2 for all j . The Hasse-W eil b ound often used in the theory of Goppa co des from cur v es is a direct consequence: | # X ( F q ) − (1 + q ) | ≤ 2 g √ q . There is a cor resp ondingly co ncrete W eil-t yp e bound for h yp ersur faces in P m , and this c an b e used to der iv e bo unds on the num b e rs of F q -rational points in hyperpla ne sections as well. A hypersurface is said to b e nonde gener ate if it not contained in any linear subspace of P m . Theorem 7. L et X b e a sm o oth nonde gener ate hyp ersu rfac e of de gr e e s in P m , m ≥ 2 . Then (8) | # X ( F q ) − ( q m − 1 + · · · + q + 1 ) | ≤ b ( s ) q ( m − 1) / 2 , wher e b ( s ) = s − 1 s (( s − 1) m − ( − 1 ) m ) is the midd le Betti numb er of a smo oth hy- p ersurfac e of de gr e e s when m is even, and one less than that numb er when m is o dd. The inequality (8) follows from the shap e of the cohomolo gy groups H i ( X ) of a smo oth hypersur fa ce in P m , which (by the Lefschetz hype rplane theorem and Poincar´ e dualit y) lo ok like the cor respo nding g roups for P m − 1 , except pos sibly in the middle dimension i = m − 1. Example 3 . If m = 2 and X is a smoo th curve o f degree s in P 2 , then b ( s ) = s − 1 s (( s − 1) 2 − 1) = ( s − 1)( s − 2) = 2 g ( X ) as exp e cted. In o rder to obtain long co des ov er F q , the maximal curves , that is, curves atta ining the maxim um # X ( F q ) from (8), hav e bee n e specially in tensively studied. F or instance, when q = r 2 , the Hermitian cur v e of degr ee s = r + 1 o ver F r 2 , X = V ( x r +1 0 + x r +1 1 + x r +1 2 ), has # X ( F r 2 ) = r 3 + 1 = 1 + r 2 + r ( r − 1) r . ♦ Example 4. When m = 3 and q = r 2 , the a na logous Hermitian su rfac es X = V ( x r +1 0 + x r +1 1 + x r +1 2 + x r +1 3 ) also a tta in the upp er b ound fr om (8), which reads # X ( F r 2 ) ≤ 1 + r 2 + r 4 + r r + 1 ( r 3 + 1) r 2 = ( r 2 + 1)( r 3 + 1) . The Hermitian surfac e co n tains this many distinct F r 2 -rational p oin ts because, for instance, it is pos s ible to take the deﬁning equation to the aﬃne form y r 1 + y 1 = y r +1 2 + y r +1 3 by a linea r change of coo rdinates that puts a plane tangent to the surface a s the plane at inﬁnit y . Then ther e a re r 5 aﬃne F r 2 -rational po in ts ( r for each pair ( y 2 , y 3 ) ∈ ( F r 2 ) 2 ). There are a lso ( r + 1) r 2 + 1 rationa l points at inﬁnity since the int e rsection of the surface with each of its tangent planes at a n F r 2 -rational point is the union of r + 1 concurrent lines in that plane. This yields r 5 + ( r + 1 ) r 2 + 1 = ( r 3 + 1)( r 2 + 1) points as cla imed. ♦ The following result o f Lac haud app ears in [37]. 10 JOHN B. LITTLE Theorem 8. L et X b e a smo oth nonde gener ate hyp ersurfac e of de gr e e s in P m for m ≥ 3 . L et H = V ( f ) for a line ar form in F q [ x 0 , . . . , x m ] , and let X H denote the interse ction X ∩ H (with t he r e duc e d scheme structur e). Then (9) | # X H ( F q ) − ( q m − 2 + · · · + q + 1) | ≤ ( s − 1) m − 1 q ( m − 1) / 2 , and (10) | q # X H ( F q ) − # X ( F q ) | ≤ ( s − 1) m − 1 ( q + s − 1) q ( m − 1) / 2 . These bo unds are pr oved b y comparing the cohomolog y of X and X H , taking into account p o ssible singula rities of X H . F or a pro of, see Corolla ry 4.6 and preceding results of [3 7]. When S is the full set o f F q -rational points on X , so n = # S for the C ( X ; S ) co de a nd H is a gener al hyperplane, these imply the following bounds on #( H ∩ S ). (10) implies (11)     ( n − #( H ∩ S )) − ( q − 1) q n     ≤ ( s − 1) m − 1 ( q + s − 1) q ( m − 1) / 2 and (12)   ( n − #( H ∩ S )) − q m − 2   ≤ s ( s − 1) m − 1 q ( m − 1) / 2 . These, toge ther with (9), give universally applicable low er b ounds on d by applying (4). As is p erhaps to b e exp ected, it is often po ssible to der iv e tighter bo unds in sp eciﬁc cases by taking the proper ties of X into acco un t. 4. Examples This sectio n w ill consider co des pro duced according to the constructions fr om Section 2 fro m v ar io us sp ecial cla sses o f v arie ties . The particular v a r ieties us ed here are all examples o f v a rieties with many rational po in ts ov er ﬁnite ﬁelds F q . The examples are ordered according to the alg ebraic geometr ic prer e quisites needed for the constr uction. 4.1. Quadrics. First c onsider the C ( X ) co des from qua dric hyper surfaces X = V ( f ) for homogeneous f o f degr ee 2 in F q [ x 0 , . . . , x m ]. The following statements are proved, for instance , in Chapter 22 of [30]. Up to pro jective equiv ale nc e ov er F q , such X are co mpletely describ ed by a p ositive integer called the r ank and a second integer called the char acter , which takes v alues in the ﬁnite set { 0 , 1 , 2 } . The rank, denoted ρ , can b e describ ed a s the minimum n umber of v aria bles needed to express f after a linear c hang e of co ordinates in P m . X is said to b e nonde gener ate if ρ = m + 1 . Nondegene r ate quadric s ar e alwa ys smo oth v ar ieties. Dege ne r ate quadrics are s ing ular, but they ar e cones ov er nondegenerate quadrics in a linear subspace of P m . Hence in principle it suﬃces to study nondegener ate quadrics and we will co nsider o nly that cas e here. The character , denoted w , is most easily describ ed by c o nsidering a ﬁnite set of p ossible nor mal forms for f . If m is even, then every nondegenerate quadric can be taken to the form x 2 0 + x 1 x 2 + x 3 x 4 + · · · + x m − 1 x m . V ( f ) is called a p ar ab olic quadric in this case, and the character w is deﬁned to b e 1. CODES FROM HIGHER DIMENSIONAL V ARIETIES 11 On the other hand, if m is o dd, there are tw o distinct pos sible forms: x 0 x 1 + x 2 x 3 + · · · + x m − 1 x m or q ( x 0 , x 1 ) + x 2 x 3 + · · · + x m − 1 x m . In the ﬁrst cas e , V ( f ) is ca lled a hyp erb olic quadric and w = 2. In the seco nd, q ( x 0 , x 1 ) is a q uadratic form in tw o v aria ble s which can b e further reduced to slightly diﬀerent norma l forms dep e nding on whether q is even or o dd. F or bo th even and odd q , in the second case, V ( f ) is ca lled a el liptic qua dric and w = 0. Theorem 9. L et X b e a nonde gener ate quadric in P m with char acter w . Then # X ( F q ) = ( q ( m +1 − w ) / 2 + 1)( q ( m − 1+ w ) / 2 − 1) q − 1 = q m − 1 + · · · + q + 1 + ( w − 1) q ( m − 1) / 2 . In particular, this result sa ys that h yp e rbolic and pa rabo lic quadrics attain the upper bound from (8) w ith s = 2, and elliptic quadrics attain the low er b ound. Because ea c h linea r section o f X is also a quadric in a lower-dimensional spa ce, Theorem 9 can b e used to determine the full weight distributions of the C ( X ) codes. In particular, Theorem 10. The C ( X ) c o de fr om a s m o oth quadric X in P m has n given in The or em 9 , k = m + 1 and (13) d =      q m − 1 if w = 2 q m − 1 − q ( m − 2) / 2 if w = 1 q m − 1 − q ( m − 1) / 2 if w = 0 . F or instance, if m is even, so w = 1 (the parab olic c a se), the h yp erplane section of X con taining the mo st F q -rational points will be a hyperb olic sec tion and d is as ab o ve. When w = 2 (for example, for co des from hyperb olic quadric s in P 3 ), the minim um w eig h t co dew o rds come from h yp erplane sections tha t ar e degenerate quadrics. The same sort of reaso ning ha s also be used by Nogin and W an to deter mine the complete hierarch y of generalized Hamming weigh ts d 1 ( C ( X )) , . . . , d k ( C ( X )). The results a r e somewhat intricate to state, though, so we refer the interested rea de r to the ar ticle s [5 6, 43] and the notes in Section 9. F or the C h ( X ) co des with h ≥ 2, the dimension can be estimated using (3 ), where I X ( h ) ≃ O P m ( h − 2). This yields k ≤  m + h h  −  m + h − 2 h − 2  . 4.2. Hermitian hy p ersurfaces. F or the C ( X ) co des constructed from the Her- mitian surfa ces of Example 4 with q = r 2 , (9) gives d ≥ ( r 2 + 1)( r 3 + 1) − ( r 2 + 1 + r 4 ) = r 5 − r 4 + r 3 . How ever, closer examination of the hyperplane sections o f the Hermitian s urface yields the fo llo wing statemen t. Theorem 11. L et X = V ( x r +1 0 + x r +1 1 + x r +1 2 + x r +1 3 ) b e the Hermitian su rfac e over F r 2 . The C ( X ) c o de on S = X ( F r 2 ) has p ar ameters  ( r 2 + 1)( r 3 + 1) , 4 , r 5  . 12 JOHN B. LITTLE Pr o of. Every F r 2 -rational plane in P 3 int e rsects X e ither in a Hermitian curve containing r 3 + 1 p o in ts ov er F r 2 , or else in r + 1 concurrent lines containing ( r + 1) r 2 + 1 points. Hence b y (4), d = n − (( r + 1) r 2 + 1) = r 5 .  The C h ( X ) codes with h > 1 are mor e subtle here. Theorem 12. L et X and S b e as in The or em 11. If h < r + 1 , t he C h ( X ) c o de has p ar ameters  ( r 2 + 1)( r 3 + 1) ,  4 + h h  , d ≥ n − h ( r + 1)( r 2 + 1)  . Pr o of. This bound follows from Theorem 1 by the fact that if f is a fo r m of degree h , then V ( f ) ∩ X is a curve of degree δ = h ( r + 1 ) in P 3 . The hypothesis on h implies that the ev aluation mapping is injective. F or lar ger h , (3) w ould b e used to determine the dimension o f the spa ce of for ms of degree h v anishing o n the Hermitian v ariety .  An even tigh ter b ound (14) d ≥ n − ( h ( r 3 + r 2 − r ) + r + 1) has b een conjecture d b y Sø rensen for these co des in [52]. The Hermitian curve and surfac e co des can be generalized as follows. (see Chap- ter 23 o f[30]). O ver a ﬁeld of order q = r 2 , cons ide r the Hermitian h yp ersurfa ce in P m deﬁned by (15) X = V ( x r +1 0 + x r +1 1 + · · · + x r +1 m ) . The mapping F ( x ) = x r is a in volutory ﬁeld automor phism of F r 2 , analo gous to complex co njugation in C , and the homo geneous p olynomial deﬁning X is a na logous to the usual Hermitian form on C m +1 given by x 0 x 0 + · · · + x m x m . The deﬁning po lynomial o f X may b e understo o d as H ( x, x ) for the ma pping H : F m +1 r 2 × F m +1 r 2 → F r 2 given by H ( x, y ) = x 0 y r 0 + · · · + x m y r m . It is clear that H is additive in each v a riable and s atisﬁes H ( λx, y ) = λH ( x, y ) and H ( x, λy ) = λ r H ( x, y ) = F ( λ ) H ( x, y ) for the automorphism F ab ov e. Hence H is an example of what is known as a sesquiline ar form on F m +1 r 2 × F m +1 r 2 . It can b e shown tha t after a linear change of co ordinates deﬁned ov er F r 2 , an y sesquilinear H on V × V , where V is a ﬁnite-dimensional F r 2 -vector spac e, can b e expressed as (16) H ( x, y ) = x 0 y r 0 + · · · + x ℓ y r ℓ for s o me ℓ ≤ dim V . H is said to be nondegenera te if ℓ = dim V a nd degener ate otherwise. It follows that every linea r section L ∩ X of a Her mitia n hypersurfa ce is also a Hermitian v arie ty in the linear subspa ce L = P W for so me v ecto r subspace W . Moreov er, if the sectio n is degener ate (i.e. ℓ < dim W in (16)), then the section is a cone ov er a nondeg enerate Hermitian v ariety in a linear subspa ce of L . Thus, the prop erties of the co des C ( X ) from the Hermitian hyper surfaces are forma lly quite similar to (and e v en so mewhat simpler than) the prop erties of co des from CODES FROM HIGHER DIMENSIONAL V ARIETIES 13 quadrics discus sed ab ov e. The main ingredient is the follo wing statement for the nondegenera te Hermitian hype rsurfaces. Theorem 1 3 . L et X b e the nonde gener ate Hermitian hyp ersurfac e fr om (15). Then # X ( F r 2 ) = r 2 m − 2 + · · · + r 2 + 1 + b ( r + 1) r m − 1 , wher e b ( r + 1) = r r +1 ( r m − ( − 1 ) m ) . In other w or ds, for all m , the nondegenerate Hermitian h yp ersurfac e s meet the upper bound from (8) fo r a hyper surface of degree s = r + 1. Theorem 14. L et S = X ( F r 2 ) for the nonde gener ate Hermitian hyp ersurfac e X in P m . The C ( X ; S ) c o de has n given in The or em 13, k = m + 1 , and d = ( r 2 m − 1 − r m − 1 if m ≡ 0 mod 2 r 2 m − 1 if m ≡ 1 mod 2 . When m is even, the minim um w eight co dewords of the C ( X ) come from non- degenerate Hermitian v ar iet y hyperpla ne sections. On the other hand, if m is o dd, then the minimum w eight co dewords of C ( X ) co me fro m h yp erplane sections that are degenerate Hermitian v ar ieties. In this case, In both cases, the nonzero co de- words o f C ( X ) hav e only t wo distinct weights: r 2 m − 1 + ( − 1 ) m − 1 r m − 1 and r 2 m − 1 . The hierarchies of g eneralized Hamming weigh ts d r are also known for the C ( X ) co des by work of Hirschfeld, Tsfas ma n, and Vladut, [31]. The same so rt of tec h- niques used in Theor em 11 ab ov e can be applied to the C h ( X ) co des for h ≥ 2 here. How ever, muc h less is known ab out the e x act Hamming weigh ts o f these co des. 4.3. Grassmannians and ﬂag v arieties . The Gr assmannian G ( ℓ, m ) is a pr o jec- tive v ar iet y whose po in ts are in o ne-to-one cor respo ndence with the ℓ -dimensional vector subspaces of an m -dimensiona l vector spac e (or e quiv alently the ( ℓ − 1)- dimensional linear subspaces of P m − 1 ). W e very brieﬂy recall the construction. Let F denote an a lgebraic closure of F q . Given any basis B = { v 1 , . . . , v ℓ } for an ℓ -dimensional vector subspace W of F m , form the ℓ × m matrix M ( B ) with rows v i . Co nsider the determinants of the maximal square ( ℓ × ℓ ) submatrices o f M ( B ). There is o ne such maximal minor fo r each subset I ⊂ { 1 , . . . , m } with # I = ℓ , so writing p I ( W ) for the maxima l minor in the columns cor resp onding to I , the Pl¨ ucker c o or dinate ve ctor of W is the homogene o us coor dinate vector (17) p ( W ) = ( · · · : p I ( W ) : · · · ) ∈ P ( m ℓ ) − 1 , where I runs thro ugh all subsets of size ℓ in { 1 , . . . , m } . The p oin t p ( W ) is a well- deﬁned inv ar ian t of W be c ause a c ha nge of ba sis in W mult iplies the matrix M ( B ) on the left b y the c hang e of ba sis matrix, an element o f GL( ℓ, F ). All co mponents of the Pl ¨ uck er co ordina te v ecto r are m ultiplied by the determinant of the change of basis matr ix, an element of F ∗ . Hence any c hoice of basis in W yields the same po in t p ( W ) in P ( m ℓ ) − 1 . The locus of all such po in ts (for all W ) forms the Gr assmannian G ( ℓ, k ). Consider the set of W such that p I 0 ( W ) 6 = 0 , so the maximal minor with I 0 = { 1 , . . . , ℓ } is inv ertible. T he set of suc h W is o ne of the o pen s ubs e ts in the standard aﬃne co ver of G ( ℓ, m ). In the r o w-reduced echelon form of M ( B ), the entries in the co lumns 14 JOHN B. LITTLE complementary to I 0 (an ℓ × ( m − ℓ ) blo ck) are arbitrary a nd uniquely determine W . Hence dim G ( ℓ, m ) = ℓ ( m − ℓ ) . T o cons truct Grass ma nnian co des, one uses the F q -rational points of G ( ℓ, m ), which come fr o m subspa ces W deﬁned ov er F q . No gin has established the following result. Theorem 1 5 . L et S b e the set of al l the F q -r ational p oints on X = G ( ℓ, m ) . Then the C ( X ; S ) c o de (fr om line ar forms in the Pl¨ ucker c o or dinates) has p ar ameters "  m ℓ  q ,  m ℓ  , q ℓ ( m − ℓ ) # , wher e  m ℓ  q = ( q m − 1)( q m − q ) · · · ( q m − q ℓ − 1 ) ( q ℓ − 1)( q ℓ − q ) · · · ( q ℓ − q ℓ − 1 ) . Pr o of. The numerator in the for m ula for  m ℓ  q is precisely the n umber of ways of picking a list of ℓ linearly independent vectors in F m q (a basis for a W deﬁned ov er F q ). Similarly , the denomina to r is the num b er of ways of picking ℓ linearly independent v ecto r s in F ℓ q , hence the or der of the group GL( ℓ , F q ). The quotient is the num be r o f distinct ℓ -dimensional subspa ces of F m q . This shows n = # S =  m ℓ  q . Assuming d = q ℓ ( m − ℓ ) for the moment, the fact that d > 0 says the ev aluation mapping on the v ecto r space o f linear forms in P ( m ℓ ) − 1 is injective, and the formu la for k follows. Finally , w e must prove that d = q ℓ ( m − ℓ ) . The complement of the hyperplane section G ( ℓ, m ) ∩ V ( p I 0 ) contains exactly q ℓ ( m − ℓ ) F q -rational p oints of G ( ℓ, m ). Hence d ≤ q ℓ ( m − ℓ ) . The cleanest wa y to prov e that this is an equality is to use the language of exter io r algebra on F q -vector spaces, following Nogin in [44]. Let V = F m q and write e i for the sta ndard basis v ecto rs in V . The F q -rational po in ts o f the Grassmannian G ( ℓ, m ) ca n b e identiﬁed with the subset of P  V ℓ V  ≃ P ( m ℓ ) − 1 corres p onding to the c ompletely de c omp osable e le men ts of the exterior pro d- uct V ℓ V (that is, nonzero element s of the form ω = w 1 ∧ w 2 ∧ · · · ∧ w ℓ for s o me w i ∈ V that form a basis for the subspace they span). The hyp e rplanes in P  V ℓ V  corres p ond to elements of P  V ℓ V  ∗ , hence to elements o f V m − ℓ V (up to scalars ) via the nondegenera te pair ing ∧ : V m − ℓ V × V ℓ V → V m V ≃ F q . It follows that the hyper planes in P  V ℓ V  all hav e the form H ( α ) = P { ω ∈ V ℓ V : α ∧ ω = 0 } for some no nzero α ∈ V m − ℓ V . Under these identiﬁcations, each hyperpla ne V ( f ) for f a linear for m in the Pl ¨ uc ker coo rdinates corresp onds to H ( α ) for some α . F or instance, V ( p I 0 ) corr e- sp onds to H ( α 0 ) for the co mpletely decompos able element α 0 = e ℓ +1 ∧ · · · ∧ e m . CODES FROM HIGHER DIMENSIONAL V ARIETIES 15 All completely decompos able α ∈ V m − ℓ V deﬁne hyperplane sections of the Grass- mannian with the same num b er of F q -rational p oints. Call this n umber N ℓ . What m ust be prov ed is that if β ∈ V m − ℓ V is arbitra ry , then the linea r for ms f in the Pl ¨ uc ker co ordinates deﬁning the hype r plane H ( β ) satisfy wt( ev S ( f )) ≥ N ℓ . This follows by induction on ℓ using the easily c hecked fact that if e ∈ V and α ∈ V m − ℓ V , then (18) α ∧ e = 0 ⇐ ⇒ α = α ′ ∧ e for some α ′ ∈ V m − ℓ − 1 V . If ℓ = 1, there is nothing to prov e b ecause e very element of V m − 1 V is completely decomp osable. If ℓ > 1 , writing [ ℓ ] q = #GL( ℓ, F q ), wt( ev S ( f )) = # { W = Span( w 1 , . . . , w ℓ ) : β ∧ w 1 ∧ · · · ∧ w ℓ 6 = 0 } = # { ( w 1 , . . . , w ℓ ) : β ∧ w 1 ∧ · · · ∧ w ℓ 6 = 0 } / [ ℓ ] q Hence by the induction h yp othesis, if α is completely decomp osable [ ℓ ] q · w t( ev S ( f )) = X w 1 : β ∧ w 1 6 =0 # { ( w 2 , . . . , w ℓ ) : ( β ∧ w 1 ) ∧ w 2 ∧ · · · ∧ w ℓ 6 = 0 } ≥ X w 1 : β ∧ w 1 6 =0 N ℓ − 1 · [ ℓ − 1] q = N ℓ − 1 · [ ℓ − 1] q · # { w 1 : β ∧ w 1 6 = 0 } ≥ N ℓ − 1 · [ ℓ − 1] q · # { w 1 : α ∧ w 1 6 = 0 } by (1 8) = [ ℓ ] q · N ℓ .  The exterio r algebra langua ge can als o be used to say more ab out the weight distribution o f C ( G ( ℓ, m ); S ). F or instance, the n umber of minimum weigh t w or ds of this c o de is equal to the num b er of linear forms corr esponding to completely decomp osable α . This num be r is exactly q − 1 times the num b e r of F q -rational po in ts o f the dual Gr assmannian G ( m − ℓ, m ), o r ( q − 1)  m m − ℓ  q = ( q − 1)  m ℓ  q . F or further infor mation on these codes see the bibliogr aphic notes in Section 9. Co des on certain sub v ar ieties o f Gra ssmannians, the so-called Schub ert varietie s , hav e a ls o b e en studied in detail by Chen, Guerra and Vincenti, and Gho rpade and Tsfasma n. Let α = ( α 1 , . . . , α ℓ ) ∈ Z ℓ , wher e 1 ≤ α 1 ≤ · · · ≤ α ℓ ≤ m . If B = { v 1 , . . . , v m } is a ﬁxed bas is of F m q , let A i be the span of the ﬁr s t i vectors in B . Then the Sch uber t v ariety Ω α is deﬁned a s (19) Ω α = { p ( W ) ∈ G ( ℓ, m ) : dim W ∩ A α i ≥ i } . See Section 9 for some p ointers to the liter a ture here. Just as Gras smannians par ametrize linear subspaces in F m , the ﬂag varieties parametrize ﬂag s o f linear subspa ces, that is nes ted sequences of subspaces V 1 ⊂ V 2 ⊂ · · · ⊂ V s , 16 JOHN B. LITTLE where dim V i = ℓ i and 0 < ℓ 1 < ℓ 2 < . . . < ℓ s < m . The ﬂag is said to have typ e ( ℓ 1 , ℓ 2 , . . . , ℓ s ). Also set ℓ s +1 = m and ℓ 0 = 0 by conv ention. The gr oup G = GL( m, F ) ac ts on the set of ﬂags of each ﬁxed t y p e and the isotr op y subgro up of a particular ﬂag is a para bolic subgroup P conjuga te to the gro up o f blo ck upper - triangular matric e s with diagonal blo cks M r of sizes ℓ r − ℓ r − 1 for 1 ≤ r ≤ s + 1. Hence the quotient G/P , which is denoted F ( ℓ 1 , ℓ 2 , . . . , ℓ s ; m ) , classiﬁes ﬂags of t yp e ( ℓ 1 , ℓ 2 , . . . , ℓ s ). The s et G/P has the structure o f a pro jective v ariety , which can b e describ ed as follows. Each V i corres p onds to a p o in t of G ( ℓ i , m ). So the ﬂag cor r espo nds to a p oint of the pr oduct v ariety G ( ℓ 1 , m ) × · · · × G ( ℓ s , m ) a nd F ( ℓ 1 , ℓ 2 , . . . , ℓ s ; m ) is the subset o f this pro duct deﬁned b y the c o nditions V i ⊂ V i +1 for a ll i . This can be embedded in P N 1 × · · · × P N s , for N i =  m ℓ i  , b y the Pl ¨ uc ker co ordinates as in (17). Finally , the pro duct P N 1 × · · · × P N s ֒ → P N for N = ( N 1 + 1 ) · · · · · ( N s + 1 ) − 1 by another standard construction called the Se gr e map . As in the Grassmannian case, F q -rational p oints on F ( ℓ 1 , ℓ 2 , . . . , ℓ s ; m ) corr e- sp ond to ﬂags that a re deﬁned ov er F q . As an example of co des from ﬂag v ar ieties, consider the co de C ( X ; S ) from X = F (1 , m − 1; m ) (that is, the v ar iet y para metriz- ing ﬂags V 1 ⊂ V 2 consisting of a line V 1 and a hyper pla ne V 2 containing that line). In this ca s e F (1 , m − 1 ; m ) ⊂ G (1 , m ) × G ( m − 1 , m ) ≃ P m − 1 × P m − 1 ֒ → P m 2 − 1 . Theorem 16. L et S b e the set of al l the F q -r ational p oints on X = F (1 , m − 1; m ) . Then the C ( X ; S ) c o de has p ar ameters  ( q m − 1)( q m − 1 − 1) ( q − 1) 2 , m 2 − 1 , q 2 m − 3 − q m − 2  . The pro of is due to Ro dier and appea rs in [47]. The ev alua tion mapping using linear forms on P m 2 − 1 is not injectiv e in this case beca use the condition that V 1 ⊂ V 2 is ex pressed by a linear equation in the co ordinates of the Seg re embedding of P m − 1 × P m − 1 . 4.4. Blo w-ups and Del P ezzo surfaces. Conside r the surface X = P 2 . Le t (20) Y k → Y k − 1 → · · · → Y 1 → Y 0 = X, be a sequence of morphisms where for all j , π j : Y j → Y j − 1 is the blo w up of an F q - rational p oint of the surface Y j − 1 . The result will be a surface Y = Y k containing divisors E 1 , . . . , E k that are all cont racted to a point on X . E ach E j is isomor phic to P 1 , and ea c h c o n tributes q additional F q -rational p oints. Therefore # Y ( F q ) = q 2 + q + 1 + k q , which also atta ins the upper W eil b ound for a surface with the Betti num b ers of these examples . Whether this construction gives in teresting c odes dep ends very m uch on the the embedding of the sur face Y into P m (that is, on the linear ser ie s of divisors for ming the hyperplane sections). One famous family of examples of suc h surfaces are the so-called Del Pezzo surfaces. Hartshorne’s text [2 8] a nd Ma nin [40] are g o o d gener al r e ferences fo r these. By deﬁnition, a Del Pezzo surface is a surfa c e of degree m in P m on which the anticanonical line bundle K − 1 is ample. A cla ssical result in the theory of CODES FROM HIGHER DIMENSIONAL V ARIETIES 17 algebraic sur faces is that every Del Pezzo surface over a n algebra ically closed ﬁeld F is obtained either as the deg ree 2 V eronese image o f a quadric in P 3 , or as follows. Let ℓ b e o ne of the integers 0 , 1 , . . . , 6, and take p oints p 1 , . . . , p ℓ in P 2 in gener al po sition (no three co llinear, and no s ix contained in a co nic curve). The linea r system of cubic curves in P 2 containing the base p oint s { p 1 , . . . , p ℓ } gives a rational map ρ : P 2 – – → P 9 − ℓ . The image is a surface X ℓ of degree 9 − ℓ o n whic h the p oints p i blow up to exceptional divisors E i ≃ P 1 as in the co mposition of all the ma ps in (20). Since the canonical sheaf on P 2 is K ≃ O P 2 ( − 3), the an ticanonical divisors are precisely the div isors in the linear system of cubics containing { p 1 , . . . , p ℓ } . F or instance, with ℓ = 6 , X ℓ is a cubic surface in P 3 , and ev er y smo oth cubic surface is obtained by blowing up some choice of p oints p 1 , . . . , p 6 . With ℓ = 0, the surface X 0 is the degree 3 V eronese image of P 2 , a s urface of deg ree 9 in P 9 . T o get a Del P ezzo surface deﬁned ov er F q , the po in ts p i should b e F q -rational po in ts in P 2 . This means that the construction ab ov e ca n fail for certain small ﬁelds (there may not b e eno ugh p oin ts p i in g eneral p osition). It suﬃces to take q > 4 , how ever in order to construct the Del Pezzo sur faces with 0 ≤ ℓ ≤ 6. By considering the possible hyperplane sections of the Del Pezzo surface Bo- guslavsky der iv es the following result in [3]. Theorem 17. L et X ℓ b e the Del Pezzo surfac e c onstructe d as ab ove and let q > 4 . The p ar ameters of t he C ( X ℓ ) c o de ar e n = q 2 + q + 1 + ℓq , k = 1 0 − ℓ, and d given in the following table ℓ 0 1 2 3 4 5 6 d ( C ( X ℓ ) q 2 − 2 q q 2 − 2 q q 2 − 2 q q 2 − 2 q + 1 q 2 q 2 + 2 q q 2 + 4 q + 1 ∗ The case ℓ = 6 co rresp onds to the co de from a cubic surfac e in P 3 . No te the asterisk in the table ab ove. In the gener ic case, there are plane sections o f a cubic surface consisting of three lines forming a tr iangle, but no sections co nsisting o f three concurr en t lines. The tr iangle plane sections c on tain the maximum n umber of F q -rational points, namely 3 q . Hence d ( C ( X 6 )) = q 2 + 7 q + 1 − 3 q = q 2 + 4 q + 1, as claimed in this case. F or so me sp ecial conﬁguratio ns of p oint s p i , how ever, the corres p onding cubic sur face will hav e Eckar dt p oints where there is a plane section consisting of three concurrent lines. F or those surfaces, the minimum distance is q 2 + 4 q rather than q 2 + 4 q + 1. 4.5. Ruled surfaces and generalizations. A ru le d surfac e is a surface X with a mapping π : X → C to a s mo oth curve C , who se ﬁb ers ov er all p oints o f C are P 1 ’s. Moreov er, it is usually r equired that π has a section, that is, a mapping σ : C → X such that π ◦ σ is the identit y on C . F or instance, over an algebr aically closed ﬁeld, quadric surfa ces in P 3 are isomorphic to the pr oduct ruled s ur face P 1 × P 1 . F or background o n these v arieties, C ha pter V of [28] is a g oo d refer ence. Starting from a curve C and a vector bundle o f r ank 2 (that is, a lo cally free sheaf of rank 2) E o n C , the pro jective spac e bundle X = P ( E ) is a ruled surface. Conv erse ly , every ruled surfac e π : X → C is isomorphic to P ( E ) for so me loca lly free sheaf of rank 2 o n C . Given a curve C and tw o vector bundles on C , the r uled surfaces P ( E ) and P ( E ′ ) a re iso morphic if and only if E ≃ E ′ ⊗ L for some line bundle L on C . By c ho osing L appropriately , it is p ossible to make H 0 ( E ) 6 = 0 but H 0 ( E ⊗ M ) = 0 whenev er M is a line bundle on C o f negative degr ee and in this 18 JOHN B. LITTLE case we say E is normalize d . Then there is a section C 0 of X with C 2 0 = − e where e = deg( E ) is the degree of the diviso r E on C cor respo nding to the line bundle V 2 E . If E is decomp osable (a dir ect sum of t wo line bundles) and normalized, then e ≥ 0. If E is indecomp osable, then it is kno wn that − g ( C ) ≤ e ≤ 2 g ( C ) − 2, w he r e g ( C ) is the genus. Up to num erical equiv alence, each divisor D on X is D ∼ b 1 C 0 + b 2 f , where f is a ﬁb er of the mapping π and b 1 , b 2 ∈ Z . The intersection pro duct on divis ors is determined b y the relations C 2 0 = − e , C 0 · f = 1, f 2 = 0. S.H. Hansen ha s shown the following result. Theorem 18. L et π : X → C b e a n ormalize d rule d surfac e with invaria nt e ≥ 0 . L et # C ( F q ) = a , and let S b e the ful l set of F q -r ational p oints on X . L et L = O X ( b 1 C 0 + b 2 f ) . Then the C ( X , L ; S ) c o de has p ar ameters [ a ( q + 1) , dim Γ( X , L ) , d ≥ n − b 2 ( q + 1) − ( a − b 2 ) b 1 ] , (pr ovide d t hat b 2 < a and the b ound on d is p ositive). Pr o of. Let f 1 , . . . , f a be the ﬁbers of π ov er the F q -rational p oin ts of C . These ar e disjoint curves on X iso morphic to P 1 , hence con tain q + 1 F q -rational points each. Every F q -rational p o in t of X lies on one o f these lines, so n = a ( q + 1). As usual, the statement for k follows if d > 0. The es timate for d comes from the method of Theorem 3 applied to the cov ering family of cur ves f 1 , . . . , f a . In the notation o f that theorem, we hav e N = q + 1 and η = ( b 1 C 0 + b 2 f ) · f = b 1 . At most ℓ = b 2 of the ﬁb ers a r e contained in any divisor D corr e s ponding to a global s ection of O X ( b 1 C 0 + b 2 f ) since D · C 0 = ( b 1 C 0 + b 2 f ) · C 0 = − eb 1 + b 2 ≤ b 2 . The bo und on d follows immediately .  The dimension of the space o f globa l sections of L can be computed via divisors on C beca use o f g eneral facts ab out sheav es on the pro jective spa ce bundle P ( E ) (see [28], Lemma V.2.4). See the bibliographic notes in Section 9 for more infor mation ab out these co des a nd for work on co des from pro jective bundles of higher ﬁb er dimension. 5. Codes fr om Deligne-Lusztig V arieties Some of the most interesting v arieties that have b een used to produce co des by the constructions of Section 2 are the so-calle d Deligne-Lusztig varieties from representation theor y . As we will see, their description inv o lv es several of the general pro cesses on v a r ieties inv olved in the examples above. Let G b e a connected r eductiv e aﬃne algebraic g roup over the algebr aically closure F of F q , a closed subgr oup o f GL( n, F ) for so me n . W e have the q -F ro benius endomorphism F : G → G whose ﬁx ed points ar e the F q -rational p oints of G . A Bor el sub gr oup of G is a maximal connected solv able subgroup of G . A torus is a subg roup of G isomorphic to ( F ∗ ) s for some s . All Bor e l subgroups are co njugate, and ea c h maximal tor us T is contained in some Bor el subgro up. Let N ( T ) b e the normalizer o f T in G . The quo tient N ( T ) /T is a ﬁnite group called the Weyl gr oup of G . The set B of all Borel subgro ups of G can b e iden tiﬁed with the quotien t G/B for any particular B via the mapping G/B → B given b y g 7→ g − 1 B g . If w ∈ W , then the Delig ne - Lusztig v a riet y asso ciated to w can b e de s cribed as follows. Let CODES FROM HIGHER DIMENSIONAL V ARIETIES 19 B b e an F -stable Bor el subgroup, then X ( w ) = { x ∈ G : x − 1 F ( x ) ∈ B w B } /B ⊂ B . Theorem 19. L et w = s 1 · · · s n b e a minimal factorization of w into simple r eﬂe c- tions in W , the Weyl gr oup of G as ab ove. Then (1) X ( w ) is a lo c al ly close d smo oth variety of pur e dimension n . (2) The variety X ( w ) is ﬁ x e d by the action of the gr oup G F and is deﬁne d over F q δ , wher e δ is the smal lest int e ger such that F δ ﬁxes w . (3) The closur e of X ( w ) in B is the union of the X ( s i 1 · · · s i r ) such that 1 ≤ i 1 < i 2 < · · · < i r ≤ n and X ( e ) . W e refer to [5] for the classiﬁcation of reductive G in terms of Dynkin diagra ms with a ction of F . In [21], J. Ha nsen studied the Hermitian curves ov er F q 2 , the Suzuki curves o ver F 2 2 n +1 and the Ree curv es ov er F 3 2 n +1 , all well-known maximal curves, and a ll used to cons truct in teresting Goppa co des with very large automor- phism gr o ups. Hansen show ed that the underlying reaso n these par ticular curves are so rich in goo d pro perties is that they are the Deligne-Lusztig v arieties for groups G for whic h ther e is just one orbit of simple r e ﬂe c tions in the W eyl group under the actio n of F . The Hermitian curves come from g roups of type 2 A 2 , the Suzuki curves comes from the gr oups of type 2 B 2 , a nd the Ree cur v es from the groups of type 2 G 2 . It is known that there are seven cases in which there ar e tw o F -or bits in the set of reﬂec tio ns in W , so ta k ing s 1 , s 2 from the distinct or bits, the Deligne- Lusztig construction with w = s 1 s 2 leads to a lgebraic surfaces: A 2 , C 2 , G 2 , 2 A 3 , 2 A 4 , 3 D 4 , 2 F 4 . One of these ca s es is rela tiv ely unin teresting. In [4 6], Ro dier shows that the com- plete, smo oth Deligne - Lusztig v a r iet y X ( s 1 , s 2 ) fr o m the group of type A 2 is iso- morphic to the blow-up o f P 2 at all of its F q -rational p oints. F or the group of t yp e 2 A 3 , how ever, Ro dier shows that X ( s 1 , s 2 ) is isomorphic to the blow-up of the Hermitian surface in P 3 at its F q 2 -rational p o in ts. Hence as in the discussio n of the blo w-ups of P 2 ab o ve, and using Example 4, we g et a surfa ce with ( q 3 + 1)( q 2 + 1) 2 po in ts. Similarly the X ( s 1 , s 2 ) from a gr o up of t yp e 2 A 4 is isomorphic to the blow-up of the complete intersection Y of the tw o hyper surfaces 0 = x q +1 0 + x q +1 1 + · · · + x q +1 4 (21) 0 = x q 3 +1 0 + x q 3 +1 1 + · · · + x q 3 +1 4 in P 4 at the ( q 5 + 1)( q 2 + 1) F q 2 -rational p oints on tha t surface. (These are the same as the F q 2 -rational p oints o n the Hermitian 3- fold in P 4 deﬁned by the ﬁrs t equation.) It is e a sy to chec k that these p oints a re all sing ular, and in fact they blow up to Hermitian curves (not P 1 ’s) on the Deligne-Lusztig surface. Hence the Deligne-Lusztig sur face X has a very lar ge n umber of F q 2 -rational po in ts in this case, # X ( F q 2 ) = ( q 5 + 1)( q 2 + 1)( q 3 + 1) . Ro dier determines the structure and num b er of F q δ -rational p oints in the G 2 , 3 D 4 , and 2 F 4 cases as well. Interestingly enoug h, his metho d is to rea lize the Deligne- Lusztig v a rieties as certain subsets o f ﬂag v a rieties as ab o ve, where the subspaces in the ﬂag s are rela ted to each other using the F ro benius endo morphism. 20 JOHN B. LITTLE Ro dier and S.H. Hansen also discuss the pro perties of the C h ( X ) codes on these v a rieties. F or instance in [26], Hansen sho ws the following result by relating c odes on Y from (21 ) and co des on the Deligne-Lusztig surface itself. Theorem 20. L et X b e the Deligne-Lusztig surfac e of typ e 2 A 4 over the ﬁeld F q 2 . F or 1 ≤ h ≤ q 2 , ther e exist c o des over F q 2 with n = ( q 5 + 1)( q 3 + 1)( q 2 + 1) , k =  4 + h h  −  4 + h − ( q + 1) t − ( q + 1)  , and d ≥ n − hP ( q ) , wher e P ( q ) = ( q 3 + 1)( q 5 + 1) + ( q + 1 )( q 3 + 1)( q 2 − h + 1 ) . Since P ( q ) has degree 8 in q , this shows that d + k ≥ n − O ( n 4 / 5 ) with n = O ( q 10 ), some very long co des indeed! Hansen also co nsiders the co des obtained from the singular p oint s on the complete in tersection fro m (21) (that is fro m the Hermitian 3-fold). 6. Connections with Other Code Constructions In this section we p oint out some connections b etw een the construction pre- sented here and some other exa mples of algebraic geometr ic co des rela ted to higher dimensional v ar ieties in the litera ture. There is a close connec tio n b et ween the co des C ( X , L ; S ) a nd the t oric c o des co nstructed from polyto pes or fans in R s as in [22]. A toric v ariety of dimension s o ver an a lgebraically closed ﬁe ld F is a v ar iet y X cont a ining a Zariski-o pen subset isomorphic to the s -dimensional a lg ebraic to r us T ≃ ( F ∗ ) s and on which T a cts in a ma nner compatible with the multiplicativ e group structure on T . The co mbinatorial data in a fan Σ in R s enco des the gluing information needed to pro duce a nor mal to ric v a riet y X Σ from aﬃne op en subsets of the for m Spec ( F [ S σ ]) where F [ S σ ] is a semigroup alg ebra a sso ciated to the co ne σ in the fan Σ. A polyto p e P in R s determines a normal fan Σ P and line bundle L P on X Σ P . The toric c o des a r e co des C ( X , L ; S ) for X = X Σ P , L = L P and S = T ∩ F s q =  F ∗ q  s . It is not diﬃcult to s ee that toric co des ar e s -dimensional cyclic co des with c ertain other pro perties gener alizing those of Ree d- Solomon co des. The study of deco ding algorithms for one-p oint alg ebraic geometric Goppa codes has b een uniﬁed and simpliﬁed by the theor y of or der domains discus s ed in [32, 14]. The article [38] s ho ws how order domains ca n be constructed fro m many o f the higher dimensiona l v ar ieties discussed here. 7. Code Comp arisons It is instructive to co mpare co des cons tructed by the metho ds describe d here a nd the bes t currently known codes for the same n, k . W e will fo cus on the minim um distance, althoug h there are many o ther considerations to o in deciding on co des for given applica tions. All co mparisons will b e made b y means of the online tables of Markus Grassl, [18]. One initial observ atio n is that man y of the v arie ties X that w e ha ve discussed hav e so many F q -rational p oin ts that the n achieved are far b ey o nd the ranges explored to date. When no explicit co des are known, it is still p ossible to make compariso ns with general b ounds. Since the k for most of the C h ( X ) co des we have CODES FROM HIGHER DIMENSIONAL V ARIETIES 21 seen are muc h smaller than n , the Griesmer b ound yie lds some informa tion. The usual form o f the Griesmer b ound (see [33]) says that for an [ n, k , d ] co de ov er F q , n ≥ k − 1 X i =0  d q i  . Given n, k , this inequality ca n a lso be used to derive an upp er b ound on realiza ble d for [ n, k ] co des that, in a s ense, impr oves the Singleto n b ound d ≤ n − k + 1. It should b e noted, how ever, that there are ma n y pairs n, k for w hich there are no co des attaining the Griesmer upp er b ound on d . W e b egin by noting the following well-known fact. Theorem 21. The pr oje ctive R e e d-Mul ler c o des with h = 1 fr om The or em 2 attain the Griesmer upp er b oun d for al l m . This follows since n = # P m ( F q ) = q m + · · · + q + 1, d = q m , and k = m + 1. F or h > 1, how ever, the presence of r e ducible forms of degree h , which can hav e many more F q -rational zero es than irre ducible forms (see the pro of of Theorem 2), tends to reduce the minimum dis ta nce relative to other co de constructions. This is true for a ll q , although the diﬀerence shows up for sma ller h the la rger q is. F or instance, in the binary cas e, the h = 2 pro jective Reed-Muller co de with m = 5 has parameters [63 , 2 1 , 16], but there a re binary [63 , 21 , 1 8] co des kno wn by [18]. Similarly , with q = 4, the h = 2 pro jective Reed-Muller co de with m = 3 over F 4 has par ameters [85 , 10 , 4 8], but there are [8 5 , 10 , 52] co des known over F 4 by [18]. In the ca ses that hav e b een explored in detail, the gap betw een the pro jective Reed-Muller codes and the b est known co des seems to increase with m for ﬁxed h , and also for h with ﬁxed m (for the cases h < q + 1 cons idered here at least). The minim um distance for the C ( X ) codes from quadrics from (13) also tend to be relatively close to the Griesmer bound for their n, k , a lthough the b ounds grow slightly faster than the actual d a s m → ∞ and slightly better codes are known in a num b er of cases. The co des from el liptic qua dr ics ( w = 0 ) are sup erior in general to those from hyperb olic q uadrics ( w = 2) w hen m is o dd. This is an interesting indication that p erhaps the “g reedy” a pproach of ma ximizing n = # X ( F q ) do es not alwa y s yield the be st co des. F or example, o ver F 8 , the C ( X ) co de fro m a h yp erb olic q ua dric in P 3 has pa - rameters [81 , 4 , 6 4 ], but there are [81 , 4 , 6 8] co des k nown by [18]. (The Griesmer bo und in this case gives d ≤ 69 .) By w ay of con tra st, the C ( X ) c o de from a n ellip- tic quadric has parameters [65 , 4 , 5 6 ], and this is the b est p ossible by the Griesmer bo und. Similar patterns hold ov er a ll of the small ﬁelds where systematic explo- ration has been done. F or larger m , how ever, it is not alwa ys the case that C ( X ) co des from elliptic quadrics meet the Griesmer b ound, and there ar e sligh tly b etter known co des in some cas es. The C 2 ( X ) co des from quadrics seem to b e similar, at least in the case m = 3, where the results of Edoukou from [12] can b e ap- plied. O v er F 8 for instance, the C 2 ( X ) co de from a hyper bolic quadric surface has parameters [81 , 9 , 49], but there are [81 , 9 , 58] co des known b y [18]. On the other hand, the C 2 ( X ) co de from an elliptic quadric has parameter s [65 , 9 , 4 7], and this matches the be s t known d for this n, k ov er F 8 . (The tightest known upper b ound is d ≤ 50.) The Hermitian hyper s urface co des s eem to b e similar to those fr om quadr ics. The C ( X ) c o des a r e q uite go o d, c o ming quite near the Grie s mer bound. F or 22 JOHN B. LITTLE instance, the Hermitian s urface co de from Theorem 11 ov e r F 16 has par ameters [1105 , 4 , 1 024]. This is far outside the r ange of n and ﬁelds for which tables are av ailable, but b y way o f comparison, d ≤ 103 4 by the Gries mer bound. Howev er, the C 2 ( X ) codes are not as goo d, and the gap gr o ws with h . The c o des from Del Pezzo surfac e s from Theo rem 17 are interesting only for ℓ = 0 (the cas e X = P 2 ) a nd ℓ = 6 (the case o f the cubic sur fa ce in P 3 ). The int e rmediate cases are quite inferior to the b est known co des. F or the other families o f v a rieties we hav e consider ed (Grassmannians, ﬂa g v a- rieties, Deligne-Lusz tig v arieties ), o nce q or m get even mo derately large, n is so hu ge that very little is known. On the basis of r a ther limited e v idence, the Grass- mannian and ﬂa g v ar iet y co des migh t b e espec ia lly g oo d only o ver very small ﬁelds, though. F or example, the C ( X ) co de from X = G (2 , 4 ) o ver F 2 has para meters [35 , 6 , 16 ], which attains the Griesmer b ound. Over F 3 , the corresp onding Gr a ss- mannian co de has [130 , 6 , 8 1], but there a re [130 , 6 , 8 4] c o des o ver F 3 known b y [18] and the Gr iesmer b ound gives d ≤ 84 in this cas e. It is unrealis tic to exp ect every co de constructed from a v ariety of dimensio n ≥ 2 to be a w o rld-bea ter. The examples here are oﬀered as evidence that we still do not kno w how this cons truction can b est b e applied to pro duce go o d co des. 8. Conclusion The study of er ror control co des constructed fro m higher dimensiona l v arie ties is an area where it is cer tainly true that w e hav e just barely b egun feeling out the lay of the land a nd just barely s cratched the surface of wha t sho uld be p ossible. If this sur v ey of past work inspires further explo ration, then one of its goals will hav e bee n achiev ed! 9. Bibliographic N otes Se ction 1. The univ er s alit y o f the Goppa co nstruction for producing linear co des is prov ed in [45]; speciﬁcally w e are referr ing to Pellik aan, Shen, a nd v an W ee’s res ult that ev ery linear co de is we akly algebr aic-ge ometric : Given C , there exists a smooth pro jective curve X , a set S of F q -rational p oin ts on X , a nd a line bundle L = O ( G ) for some divisor G with suppor t disjoint from S , such that C is iso morphic to C ( X, L ; S ) (with no restriction on the degree of G ). Although v ery little work to date has be e n done on deco ding methods, the large groups of a utomorphisms of some of the v ar ie ties consider ed here make the p ermu - tation de c o ding paradig m a p ossibility for certain of these co des. Some work along these lines has bee n done by Kroll and Vincenti, [3 4, 35]. Se ction 2. Both for ms o f the co nstruction of co des from v arieties (Deﬁnitions 1 and 2) come fro m [54], which was the ﬁrst place wher e this idea was desc r ibed in published form. The form in Deﬁnition 2 can b e made even mo re co ncrete a nd less algebraic- geometric b y the language of pr oje ctive systems o f p oint s and their asso ciated co des. Se ction 3. Theorem 2 is taken from [3 7]. It do es not include the co des for h > q bec ause the ev aluation mapping is no longer injective in those cases, T he par ameters of the C h co des fo r h > q have been studied by Lachaud in [36] and Sørensen in [53]. The gener alized Hamming weigh ts d r for the Reed-Muller co des hav e b een CODES FROM HIGHER DIMENSIONAL V ARIETIES 23 studied by Heijnen and Pellik aan in [29]. Some ideas ab out ﬁnding go o d sub co des of the C 2 co des hav e b een pr esen ted by Brouw e r in [4]. Theorem 3, the following ex ample, and the bo und us ing Sesha dri co nstan ts in Theorem 4 are all due to S.H. Hansen a nd are taken from [26]. The res ults on b ounds for the minimum distance when S is a co mplete int e r- section c ome from [17] and that ar ticle’s bibliogr aph y gives se veral sourc es for the Cayley-Bacharach theorem and moder n generalizations. The genes is for this was the o bserv a tion that if S is a r educed co mplete intersection of tw o cubic curves in P 2 , and Γ ′ is any subset of eig h t o f the nine p oints in S , then ev er y cubic tha t contains the eigh t po in ts in Γ ′ also passes through the ninth p oint in S . Related applications to co ding theory w er e discussed by Duursma, Renteria and T apia-Recillas in [10] and J. Hans en in [23]. The theorem s tated here can als o be extended to yield a criterion for MDS co des. The W eil conjectures were orig ina lly stated in [58] and prov ed in complete gen- erality by Deligne in [9] following three decades of work by Dw o rk, Serr e, Artin, Grothendieck, V erdier, and many others. W eil’s pap er gives a diﬀerent for m for middle Betti num b e r in (8 ), but it can b e seen that his form is equiv alent to our s. The discussion of W eil-t yp e b ounds follo ws Lac haud’s present ation in [37]. B ecause of spa c e limitations and the signiﬁcantly higher prerequisites needed to work with the ℓ -a dic ´ etale cohomolog y theor y in any detail in higher co dimension, we hav e fo cused only on the a pplication of Lachaud’s results to co des from h yp ersur fa ces. The discussion in [37] is cons iderably more gener a l. Edoukou has veriﬁed Sør e nsen’s conjecture (see (1 4 )) on the Her mitian surface co des in the case h = 2 in [11]. Se ction 4. The co des from quadr ics have been int ensively studied s ince at least the 19 7 5 article [59] of W olfmann. They are e specially accessible because so m uch is known ab out the se ts o f F q -rational p oints o n quadrics as ﬁnite geo metries; see Hirschfeld and Thas, [30]. The complete hiera r c hies of ge ner alized Hamming weigh ts d r for the C ( X ) codes were determined independently by Nogin in [4 3] and W an in [56]. T o aid in comparing these diﬀere n t sources, we note that W an’s inv ariant δ is related to Hirschf e ld and Thas’s (and our) c haracter w by δ = 2 − w . The c ha r acter can also b e deﬁned b y w = 2 g − m + 3 where g is the dimension of the largest linear s ubspace of P m contained in the quadric X . Compara tively little has app eared in the literature concerning the C h ( X ) co des with h > 1 on q ua drics following the work of Aubry in [1]. O ne rece nt a rticle s tudying the C 2 ( X ) co des from quadric s in P 3 is Edoukou, [12]. Hirschfeld and Thas a lso contains a wealth of information r elated to the co des on Hermitian hypersurfaces . The para meters of the C ( X ) co des were establis hed by Chak rav arti in [6], and the gener alized Hamming weigh ts were determined in b y Hirschfeld, Ts fasman, and Vladut in [31]. Grassmannia n co des were studied ﬁrst in the binary case b y C. Ryan and K. Ry an in [48, 49, 50]. The material on Grassmannian co des presented here is tak e n from [44]. In that article, Nogin also determines the complete weigh t distribution for the co des G (2 , m ) and shows that the generalized weigh ts d r of the Grassmann co des meet the genera lized Gries mer bound when r ≤ max { ℓ, m − ℓ } + 1. More information on the gener alized w eig hts was established by Ghorpade a nd Lachaud in [15] a nd these co des are also discussed a s a sp ecial case of the co de construction from ﬂa g v a rieties by Ro dier in [47]. This article als o gives the pro of of Theo r em 16. Co des from the Schu ber t v arieties deﬁned in (19) hav e been studied in [7, 20, 16]. 24 JOHN B. LITTLE The material on Del Pezzo surface co des is taken from Boguslavsky , [3]. That article also determines the complete hierarch y o f generalized Hamming w eig h ts d r for these co des. Co des from ruled s urfaces were studied by S.H. Hansen in [26] as an example of how the b ound from Theor em 3 co uld be applied. That article also addresses the cases where the inv ariant e < 0, and presents some examples inv olv ing r uled surfaces over the Hermitian elliptic cur v e over F 4 . Co des fro m ruled s urfaces were also cons idered in Lomont’s thesis, [39]. The results for co des ov er ruled surfaces hav e b een generalized to give corresp onding results for codes on pro jective bundles P ( E ) for E of all ranks r ≥ 2 b y Nak ashima in [42]. Nak ashima also consider s co des on Grass mann, q ua dric, and Hermitian bundles in [4 1]. Other work on co des from alg ebraic s urfaces is contained in the Ph.D. theses of Lomont, [39], a nd Da vis , [8]. In addition, the unpublished pr eprin t [55] of V olo c h and Za rzar and the article [60] adopt the interesting appr oach of trying to ﬁnd go od surfaces for constructing co des by limiting the pr e s ence of reducible hyperpla ne sections throug h co n tro lling the rank of the N´ eron-Severi group. Se ction 5 . Ro dier’s a rticle [4 6] is a gold mine of informatio n and techniques for the Deligne-Lusztig surfaces and Deligne-Lusz tig v a rieties more generally . The original ar ticle of Delig ne and Lusztig and a num b er o f other w o rks devoted to this construction are refer enced in the bibliogr aph y . The P icard group and other aspects of the ﬁner str uc tur e of Deligne-Lusztig v a rieties hav e b een studied by S.H. Hansen in [24, 25, 26]. Hansen’s thesis, [24] contains c hapter s corresp onding to the other articles here. Se ction 6. A standard reference for the theor y of toric v ar ieties ov e r C is F ulton’s text, [13]; the construction generalizes to ﬁelds of characteristic p with no diﬃcult y . References [1] Y. Aubry , Reed-Muller co des asso ci ated to pro j ectiv e algebraic v arieties, in: Co ding The ory and Algebr aic Ge ometry (Pr o c e e dings, Luminy 1991) , H. Stic htenot h and M.A. Tsfasman, eds. Springer Lecture Notes in Mathematics 1518 (Springer, Berlin, 199 2), 4–17. [2] E. Ballico and C. F on tanari, The Horace metho d for err or -correcting co des, Appl. Algebr a Engr g. Comm. Comput. 17 , 135–13 9 (2006). [3] M.I. Bogusla vsky , Sections of Del Pezz o s urfaces and generalized we igh ts, Pr obl. Inf. T r ansm. 34 , 14–24 (1998). [4] A. Brouw er, Linear s pace s of quadrics and new go od codes, Bul l. Belg. Math. So c. 5 , 177-180 (1998). [5] R. Carter, Finite Gr oups of Lie T yp e (Wiley , New Y or k, 1985). [6] I.M. Chakrav arti, F amilies of codes with few distinct we igh ts fr om singular and nonsingular Hermitian v arieties and quadri cs in pro jectiv e geometries and Hadamard diﬀerence sets and designs associated with tw o-weigh t co des, in: Co ding The ory and Design The ory, I , IMA V ol. Math Appl. 20 (Springer, New Y ork, 1990), 35–50. [7] H. Chen, On the minimum distance of Sch ub er t codes, IEEE T r ans. Inform. The ory , 46 , 1535–153 8 (200 0). [8] J. Davis, Algebraic geometric co des on an ticanonical surfaces, Ph.D. thesis, Unive rsity of Nebrask a, 2007. [9] P . Deligne, La conjecture de W eil , I, Publ. Math. IHES 43 , 273–307 (1974). [10] I. Duursma, C. Ren teria and H. T apia-Recill as, Reed-Muller co des on complete in tersections, Algebr a Engr g. Comm. Comput. 1 1 , 455–46 2 (2001). [11] F. Edouk ou, Codes deﬁned by forms of degree 2 on hermitian sur faces and Sørensen’s con- jecture, Finite Fields Appl. 1 3 , 616 –627 (2007). CODES FROM HIGHER DIMENSIONAL V ARIETIES 25 [12] F. Edouk ou, Codes deﬁned by form s of degree 2 on quadric surfaces, IEEE T r ans. Inform. The ory 54 , 860–864 (2008). [13] W. F ul ton, Intr o duction to toric varieties , Annals of Mathematics Studies 131, (Princeton Unive rsity Pr ess, Princeton,1993 ). [14] O. Geil and R. P ell ik aan, On the structure of order domains, Finite Fields Appl. 8 , 369–396 (2002). [15] S. Ghorpade and G. Lac haud, Higher w eights of Grassmann codes, in: Co ding The ory, Cry- p oto gr aphy, and R elate d Ar e as, Pr o c e e dings Guanajuato 1998 , J. Buc hmann, T. Høholdt, H. Stic htenot h, H. T apia-Recillas eds. (Springer, Berli n, 2000), 122–1 31. [16] S. Ghorpade and M. Tsfasman, Sc hubert v arieties, l inear codes and en umerative combina- torics, Finite Fields Appl. 11 , 684–699 (2005 ). [17] L. Gold, J. Little and H. Sc henck, Ca yley-Bacharac h and ev aluation codes on complete in- tersections, J. Pur e Appl. A lgeb r a 19 6 , 91–99 (200 5). [18] M. Grassl , Bounds on minimum distance of linear co des, av ailable online at http://w ww.codet ables.de , accessed on 2008-02-02. [19] P . Griﬃths and J. Har ris, Principles of Al gebr aic Ge ometry (Wiley , New Y ork, 1978). [20] L. Guerra and R. Vincenti, On the l i near codes arisi ng from Sch ub ert v arieties, Des. Co des Crypto gr. 33 , 173–180 (2004). [21] J. Hansen, Deligne-Lusztig v arieties and group co des, in: Co ding The ory and Algebr aic Ge om- etry (Pr o c e e dings, Luminy 1991) , H. Stich tenoth and M.A. Tsfasman, eds. Springer Lect ure Notes in Mathematics 1518 (Springer, Berlin, 1992), 63–81. [22] J. Hansen, T or ic surfaces and er ror correcting co des, Co ding The ory, Cryp oto gr aphy, and R elate d Ar e as, Pr o c e edings Guanajuato 1998 , J. Buc hmann, T. Høholdt, H. Stich tenoth, H. T apia-Recillas eds. (Springer, Berl in, 2000), 132–1 42. [23] J. Hansen, Link age and codes on complet e inte r sections, Appl . Algebr a Engr g. Comm. Com- put. 14 , 175–185 (2003). [24] S.H. Hansen, The geometry of Deligne-Lusztig v ar ieties: Higher di mensional AG co des, Ph.D. thesis Univ er s it y of Aar h us, 1999. [25] S.H. Hansen, Canonical bundles of Deligne-Lusztig v arieties. Manuscripta Math. 98 363–375 (1999). [26] S.H. Hansen, Error-correcting codes from higher-dimensional v arieties, Finite Fields Appl. 7 , 530–552 (2001) . [27] S.H. Hansen, Picard groups of Deligne-Lusztig v ari eties—wi th a view tow ard higher co di- mensions, Beitr¨ age Algebr a Ge om. 43 , 9–26 (2002). [28] R. Hartshorne, Algebr aic Ge ometry (Springer, New Y ork, 1977). [29] P . Heijnen and R. Pe llik aan, Generalized Hamming weigh ts of q -ary Reed Muller codes, IEEE T r ans. Inform. The ory 4 4 , 181–19 6 (1998). [30] J.W.P . Hirschfeld and J.A.Thas, Gene ra l Galois Ge ometries (Oxford Univ ersi t y Press, Ox- ford, 1991). [31] J.W.P . Hirschfeld, M. Tsfasman and S.G. V l adut, The weigh t hierarc hy of higher dimensional Hermitian codes, IEEE T r ans. Inform. The ory 4 0 , 275–278 (1994 ). [32] T. H øholdt, J. v an Lin t and R. P elli k aan, Algebraic geometry codes, in: Handb o ok of Co ding The ory , W. Huﬀman and V. Pless, eds. (Elsevier, Amsterdam, 1998), 871–962. [33] W. Huﬀman and V. Pless, F undamentals of Err or-Corr ecting Co des , (Cam bridge Universit y Press, Cam br idge, 2003). [34] H.-J. Kroll and R. Vincent i, PD-sets f or the co des related to s ome classi cal v ar ieties, Discr ete Math. 301 , 89–105 (2005). [35] H.-J. Kr oll and R. Vincenti, PD-sets for binary RM-co des and the codes related to the Klein quadric and to the Sch ub ert v ariety of P G (5 , 2), Discr et e Math. 308 , 408–414 (2008). [36] G. Lachaud, The parameters of pro jective Reed-Muller codes, Discr ete Math. 81 , 217–221 (1990). [37] G. Lac haud, Num b er of points of plane sections and linear codes deﬁned on algebraic v ari- eties, in: Arithmetic, Ge ometry and Co ding The ory, Pr o c e e dings Luminy 1993 , R. Pellik ann, M. P err et, S.G. Vladut eds. (W alter de Gruyter, Berl i n, 1996) , 77–104. [38] J. Little, The ubiquit y of order domains for the construct ion of error con trol codes, A dv. Math. Communic ations 1 , 1–27 (2007) . [39] C. Lomon t, Err or c orr e cting c o des on algebr aic surfac es , Ph.D. thesis, Pur due Univ er s it y , 2003, arXiv:ma th/03091 23 . 26 JOHN B. LITTLE [40] Y u.I. Manin, Cubic F orms: Algebr a, Ge ometry, Arithmetic , (North Holland, Amsterdam, 1986). [41] T. Nak ashima, Co des on Grassmann bundles, J. Pur e Appl. Algebr a 1 99 , 235-244 (2005 ). [42] T. Nak ashima, Error-correcting codes on pr o jectiv e bundles, Finite Fields Appl. 12 , 222–231 (2006). [43] D.Y u. N ogin, Generalized Hammi ng we i gh ts of co des on multidimensional quadrics, Pr obl. Inf. T r ansm. 29 , 21– 30 (1993). [44] D.Y u. Nogin, Codes asso ciated to Grassmannians, i n: Arithmetic, Geo met ry and Co ding The ory, Pr o c ee dings Luminy 1993 , R. Pellik ann, M . Perret, S. G. Vladut eds. (W alter de Gruyter, Berlin, 1996), 145–154. [45] R. Pellik aan, B. -Z. Shen and G. v an W ee, Whic h linear codes are algebraic-geomet r ic? IEEE T r ans. Inform. The ory IT-37 , 583–60 2 (1991). [46] F. Rodier, Nombre de p oin ts des s ur faces de Deli gne et Lusztig, J. Algebr a 227 , 706–766 (2000). [47] F. Rodier, Co des f rom ﬂag v arieties o v er a ﬁnite ﬁeld, J. Pur e Appl. Algebr a 17 8 , 203–214 (2003). [48] C.T. Ry an, An application of Grassmannian v ari eties to co ding theo ry , Congr. Numer. 57 , 257–271 (1987) . [49] C.T. Ryan, Pro jective codes based on Gr assmann v arieties, Congr. Numer. 57 , 273–279 (1987). [50] C.T. Ry an and K.M. Ry an, The minimum weigh t of Grassmannian codes C ( k , n ), Discr e te Appl. M ath. 28 , 149– 156 (1990). [51] J.P . Serre, L e ttr e ` a Tsfasman , in: Journ´ ees Arithmetiques, 1989 (Lumin y , 1989), Asterisque 198-200 , 351–353 (1991). [52] A. Sørensen, Rational p oin ts on hypersurfaces, Reed-Muller codes, and algebraic-geomet ric codes, Ph.D. thesis, Aarh us, 1991. [53] A. Sørensen, Pro jective Reed-Muller codes, IEEE T r ans. Inform. The ory 37 , 1567–1576 (1991). [54] M.A. Tsfasman and S.G.Vladut, Algebr aic-ge ometric c o des (Kluw er, Dordrech t, 1991). [55] J.V olo c h and M. Zarzar, A lg e br aic ge ometric co des on surfac es , preprin t. [56] Z. W an, The weigh t hierarchies of the pro jective co des f rom nondegenerate quadrics, Des. Co des Crypto gr. 4 , 283–300 (1994). [57] V.K. W ei, Gene ralized Hamm i ng w eights for linear codes, IEE E T r ans. Inform. The ory 37 , 1412–141 8 (199 1). [58] A. W eil, N um b ers of solutions of equations i n ﬁnite ﬁelds, Bul l. A mer. Math. So c. 55 , 497–508. (1949). [59] J. W olfmann, Codes pro jectifs a deux ou trois poids asso ci´ es aux hyperquadriques d’une g ´ eom ´ etrie ﬁnie, Discr e te Math. 13 , 185–211 (1975). [60] M. Zarzar, Err or -correcting codes on low-rank surfaces, Finite Fields Appl. 13 , 727–737 (2007). Dep ar tment of Ma thematic s and Computer Science, College of the Hol y Cross, W orcester, MA 016 10 USA, litt le@mathcs.holycr oss.edu

Algebraic geometry codes from higher dimensional varieties

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment