Nonstandard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes

Nonstandard linear recurring sequence subgroups in finite fields and automorphisms of cyclic co des Henk D. L. Hollm a nn Phili ps Researc h Lab orato ries Prof. H o lstla a n 4, 5 656 A A Ei ndho v en The N et herlands email : henk. d.l. hollmann@phili ps.com No v em b er 20, 2 018 Abstract Let q = p r b e a prime pow er, and let f ( x ) = x m − σ m − 1 x m − 1 − · · · − σ 1 x − σ 0 b e an irr ed ucible p olynomial ov er the finite field GF ( q ) of size q . A zero ξ of f is called nonstanda r d (of de gr e e m ) over GF( q ) if the recurren ce relation u m = σ m − 1 u m − 1 + · · · + σ 1 u 1 + σ 0 u 0 with c haracteristic p olynomial f can generate the p o wers of ξ in a non trivial wa y , that is, with u 0 = 1 and f ( u 1 ) 6 = 0. In 2003 , Brison and Nogueira ask ed for a characte r isation of all nonstandard cases in the case m = 2, and solved this p roblem for q a prime, and later for q = p r with r ≤ 4. In this pap er, we first sh o w th at classifying nonstand ard finite field elemen ts is equiv alent to classifying those cyclic co des ov er GF( q ) generated b y a single zero that p osses extra p erm u tation automorphism s. Apart from t w o sp oradic examples of degree 11 o v er GF(2) and of degree 5 o ver GF(3), r elated to the Gola y co des, there exist t wo classes of examples of n onstandard finite field elemen ts. One of these classes (t yp e I) in v olv es irreducible p olynomials f of the form f ( x ) = x m − f 0 , and is w ell-understo o d . The other class (t yp e I I) can b e obtained fr om a p rimitiv e elemen t in some su bfield by a p ro cess that we call extension and lifting. W e will use the kno w n classification of the subgroups of PGL(2 , q ) in com b in ation with a recen t result b y Brison and Nogueira to s h o w that a nonstandard elemen t of degree tw o o ver GF( q ) necessarily is of t yp e I or t yp e I I, th u s solving completely the classification problem for the case m = 2. 1 1 In tro ductio n In a sequence of pap ers [3, 4, 5, 6, 7], Brison and Nogueira in v estigated when and how a m ultiplicativ e subgroup K of a finite field can b e generated b y a linear recurrence relation of order m w it h co efficien ts in a finite field GF( q ), with q = p r and p prime. If the recurrence relation has c haracteristic equation f ( x ) ∈ GF( q )[ x ], then suc h a subgroup is called an f -sub gr oup . In particular, they call suc h an f -sub group non-standa r d if it can b e generated in a “non- ob vious” w ay (that is, not as the sequenc e 1 , ξ , ξ 2 , . . . with ξ a zero of f ). Of particular in terest is the case where the ch aracteristic p olynomial f of the re- currence relation is irreducible. In this case, there are t wo know n t yp es of nonstan- dard f -subgroups. Here one ty p e, referred to here a s typ e I , arises from the “degen- erate” case where f is of the form f ( x ) = x m − η with η ∈ GF( q ) ∗ = G F( q ) \ { 0 } . The other t yp e, referred to here a s typ e II , can b e obtained from an f -subgroup K = GF( q m ) ∗ = GF( q m ) \ { 0 } (so with f primitive ) b y a pro cess that will b e called “lifting” and “extension” in this pap er. In the “irreducible order tw o” case where f is irreducible of degree m = 2, Brison and Nogueira ha ve sho wn that there a r e no other examples if q = p in[6] and, recen tly , if q = p r with r = 2 , 3 , 4 in [7]. In this pap er w e first sho w tha t cyclic co des of length n ov er GF( q ) generated b y a single zero ξ of degree m o ve r GF( q ) that ha v e “extra” p ermutation a utomorphisms pro vide examples of nonstan- dard f -subgroups, with f the minimal p olynomial of ξ o v er GF( q ). (In fact, w e will sho w that in the irreducible case, t hese t w o exceptional kind of ob jects are e quivalent .) As a consequenc e, w e sho w that the binary and ternary Go la y co des provide new nonstandard examples, of degrees m = 11 and m = 5, resp ectiv ely . The main results in this paper, when combine d with a result fr o m a preprin t [7] b y Brison and Nogueira, can b e used to sho w that there are no other examples in the “irreducible order tw o ” case, for an y q . T o explain our a pproac h, w e in tro duce a few definitions. An elemen t ξ in an extension field of GF ( q ) will b e called nonstandar d of de gr e e m over GF( q ) if its minimal p olynomial f o ve r G F( q ) has degree m a nd the subgroup h ξ i generated by ξ is a no nstandard f - subgroup. (It t urns out that in this case al l generators a re nonstandard, with the same degree and q -order as ξ .) It can b e shown that if f is irr e ducible of degree m o ver GF( q ), then al l f - subgroups are of the form h ξ i = { 1 , ξ , ξ 2 , . . . } , for some zero ξ of f in GF( q m ). So in order to classify nonstandard f -subgroups with f irreducible, it is sufficien t to classify nonstandard finite field elemen ts. An imp ortan t no t io n in this pa p er is the q -order ord q ( ξ ) of a n elemen t ξ in some extension of GF( q ), the smallest in teger d > 0 suc h that ξ d ∈ G F( q ). There exist tw o pro cesses, that w e call “ extension” and “lifting ” , whic h, giv en a nonstandard φ o f degree m o v er a field GF( q 0 ), can b e used to obta in a nonstandard ξ of degree m ov er an extension field GF( q ), where q = q t 0 and gcd( m, t ) = 1 , with ord q ( ξ ) = ord q 0 ( φ ) and h φ i ⊆ h ξ i . The nonstandard examples of type I I are precisely the nonstandard elemen ts o f degree m ov er GF( q ) and q -o r der ( q m 0 − 1) / ( q 0 − 1) that can b e obtained fro m a primitive elemen t of degree m o ve r GF( q 0 ) b y lift ing and e xtension. 1 No w, with eac h nonstandard finite field elemen t of degree m ov er GF( q ) and q -o rder d , w e can asso ciate a subgroup Ξ in P GL ( m, q ) whic h, in the natura l action on PG( m − 1 , q ), has an orbit of siz e d . In the case where m = 2, the pro p erties of this group Ξ together with the kno wn classification of the subgroups o f PGL(2 , q ) can b e used to sho w that Ξ is actually equal to some subgroup PGL(2 , q 0 ) or PSL (2 , q 0 ) of PG L ( 2 , q ), so that d = q 0 + 1, where q = q t 0 with t o dd. Using this, w e construct a nonstandard elemen t φ of degree tw o o ve r GF( q 0 ), of q 0 -order q 0 + 1, from whic h ξ can b e obtained b y lifting and extension. No w a recent result from Brison and Nogueira [7] states that if φ is nonstandard of degree t wo ov er GF( q 0 ) and has q 0 -order q 0 + 1, then φ mus t b e primitive . As a consequenc e, in the ab ov e situation, w e can conclude tha t the nonstandard ξ is an know n example, of the second ty p e. The con tents of this pap er are as follows . In Section 2, w e first in tro duce t he problem in more detail. W e disc uss some w ell-kno wn facts concerning linear recurrence relations and linear recurring sequences , and use these to redefine t he notion of nonstandard finite field elemen ts in terms of linearized p olynomials (or q -p o lynomials). W e describe the calls of examples of t yp e I, and w e sho w that, with a few exceptions, a primitiv e elemen t is also nonstandard. In Section 3, w e sho w that the classification problem f o r nonstandard finite field el- emen ts is in fact equiv alen t to the problem of classifying the cyclic co des with a single defining zero that hav e “extra” p ermutation automorphisms. The metho ds of lifting and extension to obta in new nonstandard elemen ts f r om old ones are in tro duced in Section 4. W e illustrat e these tec hniques by cons t ructing a class of examples referred to as ty p e I I, from primitiv e elemen ts in a subfield. In Section 5, w e first use the c omp anion matrix of an irreducible p olynomial f of degree m ov er a field GF( q ) to sho w that the q - order of a zero ξ of f actually equals the r es tricte d p erio d o f f . Then, if ξ is also nonstandard, the companion matrix a nd another matrix, considered as elemen ts of PGL( m, q ), generate a subgroup Ξ of PGL( m, q ) that has an orbit of size d on PG( m − 1 , q ). In the remainder of the pap er, w e in ves tig ate this group Ξ in the case where m = 2. First, in Section 6 w e consider the case of small q -or der 3, 4, or 5. Then, in Section 7 w e use these results together with the known classification of subgroups of PGL(2 , q ) to show that Ξ is a subgroup PGL(2 , q 0 ) of PSL(2 , q 0 ), whe re q = q t 0 with t o dd, and the q -order d equals q 0 + 1 . Finally , w e establish the existe nce of a nonstandard φ of degree tw o o ve r GF( q 0 ), with q 0 -order q 0 + 1, fro m whic h the original nonstandard ξ can b e obtained b y lifting and extension. Now a recen t result b y Brison and Nogueira [7] states that a nonstandard elemen t φ of degree tw o o ver GF( q 0 ) and with q 0 -order q 0 + 1 is nec essarily primitiv e in GF( q 2 0 ), that is, has or der q 2 0 − 1; as a consequence, ξ must b e of t yp e I I. 2 2 Preliminaries Let F b e a fie ld. W e w ill write F ∗ = F \ { 0 } to denote the nonzero elemen ts in F . The collection of p olynomials in x with co efficien ts in F will b e denoted b y F [ x ]. Consider the (homogeneous linear) recurrence relatio n u k = σ m − 1 u k − 1 + · · · + σ 1 u k − m +1 + σ 0 u k − m , (1) where σ 0 ∈ F ∗ and σ 1 , . . . , σ m − 1 ∈ F . F or later use, w e define σ m = − 1. Suc h a r ecurrence relation generates for an y given sequen ce u 0 , . . . , u m − 1 in an extension field L ⊇ F of F an ( m th or der ) (homogeneous) line ar r e curring se quenc e u = u ( u 0 , . . . , u m − 1 ) in L . The (monic) p olynomial f ( x ) = x m − σ m − 1 x m − 1 − · · · − σ 1 x − σ 0 (2) in F [ x ] is called the char acteristic p olynomi a l of the recurren t relation (1); it has degree deg( f ) = m . W e will sometimes refer to a sequence u = { u k } k ≥ 0 satisfying a recurrence relation (1) with c har acteristic p olynomial f as a n f -se quenc e . F or later use, w e state some crucial fa cts concerning linear recurring sequenc es that w e need later on. T o this end, w e need a few definitions. A p erio d o f a linear recurring sequence u is a p ositiv e integer n for whic h u k + n = u k holds for all k ≥ 0; the smallest suc h n um b er is called the smal lest p erio d of the sequence, and will b e denoted b y p er( u ). The order ord( f ) of a p olynomial f is the smallest p ositiv e integer N for whic h f ( x ) divides x N − 1; if no such N exists then w e define ord( f ) = ∞ . If ξ is a no nzero elemen t in some extension L of F , then we write h ξ i = { 1 , ξ , ξ 2 , . . . } (3) to denote t he (m ultiplicativ e) group g enerated b y ξ . The order o r d( ξ ) is the smallest p ositiv e in t eger n ≥ 0 fo r whic h ξ n = 1; if no suc h n exists then ord( ξ ) = ∞ . So w e hav e that ord( ξ ) = |h ξ i| . Theorem 2.1 L et L ⊇ F b e fiel d s, and let u = u 0 , u 1 , . . . , b e a line ar r e curring se quenc e in L satisfying a r e curr enc e r elation (1) with char acteristic p olynomia l f as in (2). S up- p ose that f in F [ x ] , with σ 0 6 = 0 , and let ord( f ) < ∞ . T h en per ( u ) | or d ( f ) . Mor e over, if f has m distinct zer os ξ 0 , . . . , ξ m − 1 , then we h a ve the fol lowing. (i) Th e or der ord( f ) satisfy ord( f ) = lcm(ord( ξ i ) | i = 0 , . . . , m − 1) ; m or e over, if e ach zer o ξ i of f has the same or der n , then n = ord( f ) = p er( u ) for e ach solution u of (1) with ( u 0 , . . . , u m − 1 ) 6 = (0 , . . . , 0 ) . (ii) Supp ose that L c ontains al l these distinct zer os of f . Then eve ry s olution u of (1) in L c an b e written uniquely as u k = L 0 ξ k 1 + · · · + L m − 1 ξ k m − 1 (for al l k ≥ 0 ) with L 0 , . . . L m − 1 ∈ L . 3 Pro of: F or completeness ’ sak e, we sk etch a quic k pro of. First, if u is a s olution of (1), then since σ 0 6 = 0 w e may assume u k to b e defined for al l in tegers k , and f ( x ) ∞ X k = −∞ u − k x k = 0 . If q ( x ) f ( x ) = x N − 1, then m ultiplying b oth sides of the ab o ve relation b y q ( x ) immediately sho ws tha t N is a p erio d of u ; hence p er( u ) | N . Next, if n = p er( u ), then, writing u ∗ ( x ) = u 0 x n − 1 + u 1 x n − 2 + · · · + u n − 1 and σ m = − 1, w e ha v e that f ( x ) u ∗ ( x ) = (1 − x n ) h ( x ) , where h ( x ) = m − 1 X j = 0 m − 1 − j X i =0 σ i + j +1 u i x j is a polynomial of degree at mo st m − 1. So if ord( ξ i ) = N = p er( f ) > n ho lds for all i , then ξ n i 6 = 0, hence h ( ξ i ) = 0, for a ll i = 0 , . . . , m − 1, which is not p ossible since h has degree less than m . Finally , if ξ is a zero of f in L , t hen u k = ξ k defines a solution to (1) in L . So ob viously eac h L -linear com bination of these m solutions is also a solutio n in L . Now the statemen t follo ws f r o m the observ ation that L 0 , . . . , L m − 1 can b e uniquely determined from ( u 0 , . . . , u m − 1 ) in terms of a V andermonde matrix o ver L . ✷ Remark 2.2 In [3], it was claime d that if f is irr e ducible over a fi n ite field F , then as a c onse quenc e o f [14], The or em 8.28 , e ach n onzer o so l ution of the r e curr enc e r e l a tion (1) with char acteristic e quation f has smal lest p erio d o rd( f ) . However, the cite d the or em only claims this to hold for solutions in F . The ab ove pr o of, which, by the way, involves the same elements as the p r o of of Ther o em 8.28 in [14], shows that this also h o lds for solutions in an extens io n of F . In [3], a finite m ultiplicativ e subgroup K of some extension L of F is called an f - sub gr oup if it can b e generated without rep etitions by the recurrence r elat io n (1) with c haracteristic p olynomial f . That is, K is an f - subgroup if there is a choice of u 0 , . . . , u m − 1 in K suc h that the recurring seque nce u = u 0 , u 1 , . . . generated b y (1) has (smallest) p erio d | K | and K = { u 0 , . . . , u | K |− 1 } . Note that we ma y assume without loss of g eneralit y that u 0 = 1 b y dividing all members of the sequence u by u 0 , if necessary . W e say that K is a n m th or der line ar r e curring se quenc e sub gr oup if there is an f of degree m as in (2) with σ 0 6 = 0 suc h that K is an f -subgro up. F or later use, w e note the follow ing. F or a ll fields L , a finite subgroup K of L ∗ is necessarily cyclic , see for example []. So if | K | = n , then K consists precisely of the n solutions o f the eq ua tion x n − 1, whic h m ust therefore all be dis tinct. W e conclude that for a giv en field F , there ex ists a unique subgro up K of order n in some extension of F 4 precisely whe n the c haracteristic c har( F ) of F satisfies ( n, c har( F )) = 1. In that case K is cyclic, of the form K = h ξ i , where ξ is a primitiv e n th ro ot of unit y in an e xtension L of F . If ξ a zero of a p olynomial f ( x ) ∈ F [ x ], then the sequence u k = ξ k satisfies the recurrence relation (1 ) , and hence K = h ξ i is an f -subgroup. In [3], an f -subgroup K with f of degree m = deg( f ) = 2 was called nonstandar d if K can b e generated by a solution u of (1) with smallest p erio d | K | for whic h u 0 = 1 and u 1 is not a zero of f . Here, w e extend this to the case of general degree, b y calling an f -subgroup K = h ξ i nonstandar d if K can b e generated b y a solution u of (1) with smallest p erio d | K | for whic h u 0 = 1 and ( u 0 , u 1 , . . . , u m − 1 ) 6 = (1 , ξ , . . . , ξ m − 1 ) for all zeros ξ of f . An f -subgroup that is not nonstandard is called standar d . Theorem 2.3 If f is irr e ducible over F , if o rd( f ) < ∞ , and if f has no mult iple zer os, then e ach f -sub gr oup K in an extension L o f F is of the form K = h ξ i , for some zer o ξ of f in L . Pro of: By Theorem 2 .1, under these assumptions all nonzero solutions u of the recurrence relation with c haracteristic equation f hav e smallest p erio d n = ord( f ) = ord( ξ ), for an y zero ξ of f . So an f - subgroup is cyclic of size n , and since it is uniqu e it mus t b e equal to the group h ξ i . ✷ Remark 2.4 As state d in [3], even when f is not irr e ducib l e , no f -sub gr o up is known that is not of the form h ξ i for a zer o ξ of f , but it has n ot b e en pr ove d that this m ust hold in gener al. In this paper, w e will b e in terested in nonstandard f -subgroups. In view of the pre- ceeding remarks and observ ations, it seems reasonable to somewhat restrict our atten tion. F rom no w on, we will assume that F is a finite field GF( q ) with q = p r and p prime, and that f is ir reducible o ver F. If f is irreducible of degree m ov er GF( q ), then f ha s zero es ξ , ξ q , . . . , ξ q m − 1 , (4) for some ξ ∈ F q m , of order n = ord( f ) dividing q m − 1. Of course all ze r o s of f generate the same gr o up h ξ i , whic h is an f -subgroup. So in view of Theorem 2 .3, the follo wing definition mak es sense. W e will sa y that an elemen t ξ in some extension of GF( q ) is nonstandar d of de g r e e m over GF( q ) and or der n = ord( ξ ) if the minimal po lynomial f ( x ) of ξ ov er GF( q ) has degree m and h ξ i is a nonstandard f - subgroup, of order (size) n . With this definition, the clasification problem of nonstandard elemen ts o ve r GF( q ) is equiv alen t to the classification of nonstandard f - subgroups with f irreducible o v er GF( q ). W e will show later that if f is irreducible of degree m ov er GF( q ) and K is a nonstandard f -subgroup o f order n , then al l elemen ts of order n in K (that is, all generators of K ) 5 are nonstandard of degree m o v er GF( q ) (but with differen t minimal p o lynomials). Or, stated differen tly , if K is a nonstandard f -subgroup with f irreducible ov er GF( q ), then K is a nonstandard g -subgroup for al l minimal p o lynomials g o v er GF( q ) of generators of K . The solutions of a recurrence relation for whic h the characteristic equation is irre- ducible can b e de scrib ed in terms of line arize d p olynomia ls , see, e.g., [14], Chapter 8. A q -p olynomial of q -order m ov er an extension field L of GF( q ) is a p olynomial of the form L ( x ) = L 0 x + L 1 x q + · · · + L m − 1 x q m − 1 with co efficien ts L j in L f or j = 0 , . . . , m − 1, with L m − 1 ∈ L ∗ . Sometimes, a q -p olynomial is also referred to as a line arize d p olynomial, if the v alue of q is eviden t fro m the con t ext. Note that suc h a p olynomial is F q -line ar , that is, L ( ax + by ) = aL ( x ) + bL ( y ) for all a, b ∈ F q . W e will call a q -p olynomial nonstandar d if it is no t of the fo rm L ( x ) = cx q j for some constan t c a nd some nonnegativ e integer j , and standar d otherwise. Theorem 2.5 L et ξ ∈ GF( q m ) have minimal p olynomi a l f ( x ) over G F( q ) a s in (2). (i) A se quenc e u = { u k } k ≥ 0 in GF( q m ) satisfies the line ar r e curring r elation (1) wi th char acteristic p olynom i a l f if and only if ther e exists a q -p olynomial L ( x ) of q -or der m over G F( q m ) such that u k = L ( ξ k ) for al l k ≥ 0 . (ii) We have that ξ is nonstand ar d of d e gr e e m over GF( q ) if and o n ly if ther e exists a nonstandar d q -p olynomial L ( x ) of q -o r der m such that L ( h ξ i ) = h ξ i . Pro of: (i) Since f is the minimal p olynomial of ξ o v er GF( q ), w e ha ve that f is irre- ducible, with distinct zeros ξ , ξ q , . . . , ξ q m − 1 , all in GF( q m ). So if a se quence u = { u k } k ≥ 0 in GF( q m ) satisfies the recurrency (1) with c haracteristic p olynomial f , then according to Theorem 2.1, there are L 0 , . . . , L m − 1 in GF( q m ) suc h that u k = L 0 ξ k + L 1 ξ k q + · · · + L m − 1 ξ k q m − 1 for all k ≥ 0. So if we let L ( x ) = L 0 x + L 1 x q + · · · + L m − 1 x q m − 1 , then L ( x ) is a q -p olynomial of q -or der m ov er GF( q m ) for whic h u k = L ( ξ k ) for all k ≥ 0. (ii) F rom part (i), we seen that the subgroup h ξ i is generated by a solution u in GF( q m ) of the recurrence relation with c haracteristic p olynomial f if and only if the q -p olynomial L ( x ) o f q - order m corrsp onding to this solution u satisfies L ( h ξ i ) = h ξ i . (Note that this can only happ en if L is q -p olynomial over G F( q m ).) No w since L is q -linear and since 1 = ξ 0 , ξ , . . . , ξ m − 1 constitute a basis for GF ( q m ) ov er GF( q ), the co efficien ts L 0 , . . . , L m − 1 of L are uniquely determined b y the images L (1) , L ( ξ ) , . . . , L ( ξ m − 1 ). By replacing L ( x ) b y L ′ ( x ) = L ( x ) /L (1) if necess ar y w e may assume that L (1) = 1. ( No te that this do es not c ha nge the “standardness” o f the q -p olynomial at hand.) Then the standar d q - p olynomials L ( x ) = x q j , j = 0 , . . . , m − 1, are precisely the q - p olynomials that result in a 6 “standard” generation of the f - subgroup h ξ i where ( u 0 , . . . , u m − 1 ) = (1 , ξ q j , . . . , ξ ( m − 1) q j ). ✷ Next, w e will discuss t w o nonstandard examples. Note that there are no nonstandard elemen ts of degree m = 1. Example 1: The case where m > 1 and ξ ∈ F ∗ q m has order n > 4 and minimal p olynomial of the f orm f ( x ) = x m − η with η = ξ m ∈ F ∗ q with η 6 = 1. Note that if q = p r with p prime, then ( p, m ) = 1. Also, w e m ust ha v e q > 2: indeed, if q = 2, then η = 1 is the only p o ssibility , but since x m − 1 = ( x − 1)(1 + x + · · · + x m − 1 ) is reducible, this do es not o ccur. Under the ab ov e assumptions, ξ has q -order d = m , and h ξ i = { 1 , η , η 2 , . . . , η e − 1 } × { 1 , ξ , . . . , ξ m − 1 } , where e = n/m is the order of η and n is the order of ξ . Note that e > 1, since if e = 1, then η = 1 and x m − 1 is not irreducible for m > 1. No w let τ ∈ S m b e a p erm utation with τ (0) = 0, a nd de fine L ( ξ j ) = η j ξ τ ( j ) for j = 0 , . . . , m − 1. Finally , e xtend L b y F q -linearit y to all o f F q m . Since 1 , ξ , . . . , ξ m − 1 constitute a basis of F q m o ve r F q and since τ is assumed to be a p ermutation, L is w ell- defined b y F q -linearit y , and nonsingular on F q m . Hence, since L h ξ i ⊆ h ξ i b y definition, w e a ctually ha v e equalit y here. There are precisely e m − 1 ( m − 1)! p ossible q -p olynomials L with L (1) = 1 and precisely m forbiden (standard) ones. Hence if e = 2, m ≥ 3 or e ≥ 3, m ≥ 2, then some L is nonstandard. This condition holds precisely when m ≥ 2 , e > 1, and n > 4. In particular, it is easily ve r ified that there is an example o f degree 2 o v er F q with order n and q -o r der 2 if and only if n = 2 e > 4 and b oth q and ( q − 1) /e are o dd. W e will refer to suc h examples as exam ples of typ e I . ✷ Example 2: If m > 2 or m = 2 , q > 2, then a primitive elemen t of F q m is nonstandard o ve r F q . This is the case where ξ ∈ F q m has order n = q m − 1, so that h ξ i = F ∗ q m , where F ∗ q m = F q m \ { 0 } . Indeed, in that case any q -p olynomial L ∈ F q m [ x ] that is nonsingular on F q m will fix F ∗ q m as a set, so is nonstandard p olynomial for ξ except when of the form ξ c x q j for some c ∈ { 0 , . . . , q m − 2 } and some j ∈ { 0 , 1 , . . . , m − 1 } . Here, L is called nonsingular if the asso ciated F q -linear map L on F q m is nonsingular; equiv a len tly , if L ( x ) 6 = 0 for x ∈ F ∗ q m . Note that the requiremen t that L is nonsingular is necesary and sufficien t fo r L to act as a p erm utat io n on F ∗ q m . No w there a re precise ly ( q m − 1)( q m − q ) · · · ( q m − q m − 1 ) nonsingular F q -linear maps on F q m , whic h are all of the for m of a q - p olynomial in F q m [ x ]. Precisely m ( q m − 1) of these are “f orbidden”, but a ll others prov ide nonstandard q -p olynomials. It is easily sho wn that for in tegers m , q ≥ 2, w e hav e ( q m − 1)( q m − q ) · · · ( q m − q m − 1 ) > m ( q m − 1) except when m = 2 and q = 2. 7 W e will see later that primitiv e elemen ts are particular cases of a class of examples referred to as typ e II examples. ✷ 3 Automorphi sms of cyclic co des In this section, w e will show that the classification problem of nonstandard elemen ts ov er GF( q ) is equiv a len t to the problem of determining whic h cyclic co des o ve r G F( q ) defined b y a single zero hav e “extra” automorphisms. W e b egin b y a brief in tr o duction to cyclic co des. F or more details, see e.g. [15 ]. W e will denote by S n the collection of all p erm utations o n { 0 , 1 , . . . , n − 1 } . In what follo ws, w e will slightly a buse notation and use the same sym b ol π to denote b oth a p erm utation from S n and the induced p erm utation on the n -dimensional v ectorspace GF( q ) n giv en b y π : c 7→ c π = ( c π (0) , . . . , c π ( n − 1) ) . A cyclic c o de of length n over GF( q ) is a GF( q )-linear subspace of GF( q ) n closed under the map σ : ( c 0 , . . . , c n − 1 ) 7→ ( c n − 1 , c 0 , . . . , c n − 2 ) . This map, as w ell as the underlying p erm utation σ : i 7→ i − 1 mo d n, are b oth referred to as a cyclic shift . In w ha t follow s, we will iden tif y a v ector c = ( c 0 , . . . , c n − 1 ) ∈ G F( q ) n with it s asso ci- ated p olynomial c ( x ) = n − 1 X i =0 c i x i in GF( q )[ x ] mo d x n − 1. Note that the cyclic shift c σ of a v ector c has corresponding p olynomial c σ ( x ) = xc ( x ); so m ultiplication b y x in GF( q )[ x ] mo d x n − 1 cor r espond to a cyclic shift. Let n | q m − 1, and let Z ⊆ F ∗ q m b e a collection of field elemen ts of order dividing n , so that α n = 1 holds for all α ∈ Z . The cyclic c o de of length n over GF( q ) with defining zer o es Z is the collection C = C ( n, q , Z ) of all c = ( c 0 , . . . , c n − 1 ) ∈ GF( q ) n for whic h c ( α ) = n − 1 X i =0 c i α i = 0 holds for all α ∈ Z . W e r efer to an elemen t c ∈ C as a c o de wo r d . Note that if c ( x ) is in C , then the cyclic shift xc ( x ) is again in C ; since a cyclic co de is also linear it is in fact an ide al in GF( q )[ x ] mo d x n − 1. If c ( x ) has all its co efficien ts in GF( q ), then c ( x ) q = c ( x q ). As a consequence, the co des C ( n, q , Z ) a nd C ( n, q , ¯ Z ) are equal, where ¯ Z = { z q j | z ∈ Z ; i = 0 , . . . m − 1 } . 8 A p ermutation π ∈ S n is called a p ermutation automorphism of a cyclic co de C ⊆ GF( q ) n if for all co de w or ds c = ( c 0 , . . . , c n − 1 ) ∈ C , the p erm uted w o rd c π = ( c π (0) , . . . , c π ( n − 1) ) is again in C . Now c π ( ξ ) = n − 1 X i =0 c π ( i ) ξ i = n − 1 X j = 0 c j ξ π − 1 ( j ) , so b eside the cyclic shift σ als o the F rob enius perm utation φ : i 7→ q i mo d n is a permu- tation automorphism of a cyclic co de of length n ov er GF( q ). The next theorem pro vides a relation b etw een automorphisms π of cyclic co des a nd q -p olynomials fixing sets h ξ i . Theorem 3.1 L et ξ ∈ GF( q m ) ∗ have or der ord( ξ ) = n an d de gr e e m over GF( q ) , and let C ⊆ GF( q ) n b e the c yclic c o de C = C ( n, q , { ξ } ) of le n gth n over GF ( q ) w i th definin g zer o ξ . Then a p ermutation π ∈ S n is a p ermutation automorphism of C if and on ly if the map L : ξ i 7→ ξ π ( i ) extends t o a q -p olynomial of q -or der m over GF( q m ) . Pro of: First, suppo se that L is a q -p olynomial of q -degree m that fixes h ξ i , and let π ∈ S n b e the permutation induced b y L , that is, let π b e suc h that L ( ξ i ) = ξ π ( i ) for all i = 0 , . . . , n − 1. Then if c ∈ C , w e ha v e 0 = L (0) = L ( n − 1 X i =0 c i ξ i ) = n − 1 X i =0 c i L ( ξ i ) = n − 1 X i =0 c i ξ π ( i ) = n − 1 X j = 0 c π − 1 ( j ) ξ j that is, c π − 1 is in C . So π − 1 , a nd hence also π , is a perm uta t ion automorphism of C . Con v ersely , let π b e a p erm uta tion automorphism of C . W e define a q -p olynomial L ( x ) = L 0 x + · · · + L m − 1 x q m − 1 of q -or der m by letting L ( ξ j ) = ξ π ( j ) (5) for j = 0 , . . . , m − 1, and then extending L to all of GF( q m ) b y GF( q )- linearit y . Note that since we assumed that ξ has degree m ov er GF( q ), w e ha v e that 1 , ξ , . . . , ξ m − 1 constitute a basis for GF( q m ) ov er G F( q ), so L is uniquely determined. W e claim that now (5) holds f or al l j = 0 , . . . , n − 1. Indeed, let j ≥ m . By our assumptions on ξ , there a re a 0 , . . . , a m − 1 ∈ F q suc h that ξ j = a 0 + a 1 ξ + a 2 ξ 2 + · · · + a m − 1 ξ m − 1 . Note that then L ( ξ j ) = a 0 ξ π (0) + a 1 ξ π (1) + a 2 ξ π (2) + · · · + a m − 1 ξ π ( m − 1) . (6) No w since C is the co de with defining zero ξ , the w ord c = ( a 0 , . . . , a m − 1 , 0 , . . . , 0 , − 1 , 0 , . . . , 0) , 9 with c i =      a i , if 0 ≤ i ≤ m − 1; − 1 , if i = j ; 0 , otherwise, is in C . So by our assumption that π , and hence also π − 1 , is a p erm utation automorphism of C , the w o rd c π − 1 is also in C . Hence we hav e that 0 = n − 1 X i =0 c i ξ π ( i ) = − ξ π ( j ) + a 0 ξ π (0) + · · · + a m − 1 ξ π ( m − 1) = − ξ π ( j ) + L ( ξ j ) . W e conclude that L ( ξ j ) = ξ π ( j ) holds for al l j = 0 , . . . , n − 1, as claimed. ✷ In view of Theorem 2.5, w e immediately hav e the following consequence. Corollary 3.2 L et ξ ∈ G F( q m ) ∗ have or der ord( ξ ) = n and de gr e e m over GF( q ) . Then the cyclic c o de C ( n, q , ξ ) o f length n over GF( q ) with defining zer o ξ has “extr a p ermu- tation automorphisms”, that is, a p ermutation automorphism gr oup stric ktly larger than the gr oup h σ , φ i of or der mn gener ate d by the cyclic shift and the F r ob enius p ermutation, if and only if ξ is nonstandar d of or der n an d de gr e e m over GF( q ) . F rom the ab o v e corollary w e can obtain tw o new examples of nonstandard elemen ts. Example 3: (Binary Gola y) Let q = 2, n = 23 , and m = 11. Then n | 2 11 − 1. Let α b e primitiv e in GF(2 11 ), and let ξ = α (2 11 − 1) / 23 . Then ξ is a primitive 23-th ro ot of unit y in GF(2 11 ). The binary G olay c o d e is the binary length n = 23 co de with definin g zero ξ . It can b e sho wn that this co de has minim um distance 7 (in fact, it a p erfe c t binary 3-error- correcting co de). Its automorphism group is the Mathieu group M 23 , a simple group of order 200960, en tirely consisting o f p erm utations. As a consequence of Corollary 3.2, w e conclude that ξ is nonstandard of o rder n = 23 and degree m = 11 ov er GF (2). Its 2-order is d = 23 > m , and w e see immediately that this provides an example not of the form of the tw o k nown t yp es. ✷ Example 4: (T ernary Golay) Let q = 3, n = 11, and m = 5. Then n | 2 m − 1. Let α b e primitiv e in GF(2 5 ), and let ξ = α (2 5 − 1) / 11 . Then ξ is a primitive 1 1-th r o ot of unity in GF(2 5 ). The ternary Golay c o de is the t ernar y length n = 11 code w it h defining zero ξ . It can b e sho wn that t his co de has minim um distance 5 (in f a ct, it a p erfe ct ternary 2- error-correcting co de). Its automorphism group is tw ice the Mathieu group M 11 , a simple group of order 7920, whic h itself consists en tirely o f perm uta tions. As a consequence of Corollary 3.2, we conclude that ξ is nonstandard of or der n = 11 and degree m = 5 ov er GF (3). Its 3-order is d = 11 > m , and w e see immediately that this pro vides another example not of the form of the tw o kno wn t yp es. ✷ Examples of cyclic co des with “extra” automorphisms seem to b e quite rare. F or example, the only (binary) quadratic-residue co des with “ extra” a ut o morphisms of length less than 4000 are the (7 , 4 , 3 ) Hamming co de and t he b inary Golay co de [10]. 10 4 Extensio n and lifting F or later use, w e inv estigate when w e can conclude that a q -p olynomial L in GF ( q m )[ x ] for some φ of degree m tha t acts as a bijection on some other subgroup h ξ i of GF( q m ) ∗ is actually nonstandard for ξ . The result is as follo ws. Lemma 4.1 L et L b e a q -p olynomial in GF( q m )[ x ] , and let ξ have de gr e e m over GF( q ) . If L ( ξ i ) = ξ iq j for i = 0 , . . . , m − 1 , then L ( x ) = x q j on G F( q m ) . Pro of: If ξ has degree m o v er GF( q ), then 1 , ξ , . . . , ξ m − 1 constitute a ba sis for GF( q m ) / GF( q ), hence a q -linear map L on GF( q m ) is determined on GF( q m ) by the images on 1 , ξ , . . . , ξ m − 1 . Also, if L in in GF( q m )[ x ], then L is determined as a p olynomial b y its action on GF( q m ). ✷ W e a lso ne ed the following simple observ a tion concerning degrees. Lemma 4.2 (i) An elem ent ξ ∈ GF( q m ) of or der n has de gr e e m over GF( q ) if a n d o n ly if m is the smal lest inte ger t ≥ 1 for which n | q t − 1 . (ii) I f φ has de gr e e m over GF( q ) and ξ ∈ GF( q m ) has φ ∈ h ξ i , then ξ also has de gr e e m over G F( q ) . Pro of: (i) If n | q t − 1, then h ξ i = θ ( q t − 1) /n ⊆ GF( q t ) ∗ . (ii) If φ ∈ h ξ i , then h φ i ⊆ h ξ i , hence the order n of φ divides the order o f ξ . Now the result follows from part (i). ✷ The or der ord( ξ ) of an elemen t ξ ∈ GF( q m ) of degree m o ve r GF( q ) w a s defined a s the smallest p ositiv e in teger n for whic h ξ n = 1. W e no w define the q -or der ord q ( ξ ) as the smallest p ositiv e integer d for whic h ξ d ∈ GF( q ). The q -order is an imp ortant notion in this pap er. It is related to another no tion, the resticted p erio d, whic h w as inv estigated in [5] and pla y ed an imp orta nt role in [6] and [7]. Here, the r estricte d p erio d δ ( f ) of a p olynomial f ( x ) ∈ GF( q )[ x ] as in (2), w ith corresponding recurrence relation (1), is the first p ositiv e in teger n for whic h t he s o lutio n u = { u k } k ≥ 0 of (1) with ( u 0 , u 1 , . . . , u m − 2 , u m − 1 ) = (0 , 0 , . . . , 0 , 1 ) satisfies ( u 0 , u 1 , . . . , u m − 2 , u m − 1 ) = (0 , 0 , . . . , 0 , λ ) , for some λ ∈ GF( q ) ∗ . The next theorem states this relatio n. Theorem 4.3 The q -or der of an element ξ in an extension GF( q m ) of GF( q ) is e qual to the r estricte d p erio d of its minimal p olynomial o v er GF( q ) . W e will pro ve this theorem in Section 5. In the next theorem, we collect some imp ortant prop erties of the q -order. 11 Theorem 4.4 L et ξ ∈ GF( q ) ha ve de gr e e m ov e r G F( q ) , with or der n = ord( ξ ) and q -or der d = ord q ( ξ ) . (i) We have m ≤ d an d d | ( q m − 1) / ( q − 1 ) . (ii) We have that d = n/ ( n, q − 1) and n = de , wher e e = ( n, q − 1) satisfies ( d, ( q − 1 ) / e ) = 1 . Pro of: (i) If d = ord q ( ξ ) and ξ d = η ∈ GF( q ), the n ξ is a zero of the p olynomial x d − η in GF( q )[ x ]. Hence the minimal p olynomial o f ξ , of degree m b y o ur assumptions, divides x d − η , whence m ≤ d . The collection of all in tegers k ≥ 0 for whic h ξ k ∈ GF( q ) is an ideal, hence is of the form d Z . Now since ξ ∈ GF ( q m ) ∗ , we ha v e that ϕ = ξ ( q m − 1) / ( q − 1) satisfies ϕ q − 1 = 1, hence ϕ ∈ GF( q ). W e conclude that d | ( q m − 1) / ( q − 1). (ii)W e hav e ξ d ∈ G F( q ) if and only if ξ d ( q − 1) = 1, that is, if and only if n | d ( q − 1 ), or, equiv alently , if and only if n/ ( n, q − 1) div ides d ( q − 1) / ( n, q − 1). Since n/ ( n, q − 1) and ( q − 1) / ( n, q − 1) are relative ly prime, the latter happens if and only if n/ ( n, q − 1) divides d . If w e now write e = ( n, q − 1), then n = de and now the condition on e f ollo ws from the expression for d . ✷ W e will refer to our next theorem as the ex tens ion the or e m . It enables us to extend a nonstandard subgroup to a bigger one. Theorem 4.5 L et φ b e nonstand ar d of de gr e e m over G F( q ) . Th en e very ξ ∈ GF( q ) ∗ h φ i for which h φ i ⊆ h ξ i (so with ξ = λφ and φ = ξ i for some λ ∈ GF( q ) ∗ and inte ger i ) is again nonstanda r d of de gr e e m over GF( q ) , with the same q -o r der as φ ; mor e over, every q - p olynom ial L ( x ) of q -de gr e e m over GF( q m ) for which L ( h φ i ) = h φ i sa tisfie s L ( h ξ i ) = h ξ i ) . Pro of: W e b egin b y observing that GF( q ) ∗ h φ i is a m ultiplicativ e s ubgr o up of GF ( q m ) ∗ ; since G F( q m ) ∗ is cyclic, all its subgroups are also cyclic, and hence there exists an elemen t θ ∈ G F( q m ) ∗ suc h that GF( q ) ∗ h φ i = h θ i . No w let n = ord( φ ) and d = ord q ( φ ) denote t he order and q -order of φ , resp ectiv ely . According to Theorem 4.4, w e ha v e that d = n/ ( n, q − 1 ) . W rite k = ( q − 1) / ( n, q − 1); for later use , w e note that ( d, k ) = 1 and k | q − 1. Now GF( q ) ∗ h φ i = GF( q ) ∗ { 1 , φ, . . . , φ d − 1 } has size ( q − 1) d , hence θ has order d ( q − 1) = nk . So by Theorem 4.4 , θ has q - order ord q ( θ ) = d ( q − 1) / ( d ( q − 1) , ( q − 1)) = d = ord q ( φ ), so θ and φ hav e the same q - order. No w let L b e a q - p olynomial of q -degree m ov er GF( q m ) that fixes h φ i , sa y with L ( φ i ) = φ π ( i ) for some p erm utatio n π ∈ S n . W e claim that L ( h θ i ) = h θ i . T o see this, first note that if α ∈ GF( q ), then L ( αφ i ) = αL ( φ i ); hence L ( h θ i ) = L (G F( q ) ∗ h φ i ) ⊆ GF( q ) ∗ h φ i = h θ i . Now, suppo se that L ( αφ i ) = L ( β φ j ) for some α, β ∈ GF( q ) ∗ and some in tegers i, j . Then αφ π ( i ) = αL ( φ i ) = L ( αφ i ) = L ( β φ j ) = β L ( φ j ) = β φ π ( j ) , 12 and hence γ = α/ β = φ π ( j ) − π ( i ) ∈ GF( q ) ∗ . Therefore, L ( φ j ) = φ π ( j ) = γ φ π ( i ) = γ L ( φ i ) = L ( γ φ i ) = L ( φ π ( j ) − π ( i )+ i ) . Since L ( h φ i ) = h φ i , we conclude that φ j = φ π ( j ) − π ( i )+ i , hence γ φ i = φ π ( j ) − π ( i )+ i = ξ j , so that α φ i = β φ j . W e ha v e shown that L is one-to-one on GF( q ) ∗ h φ i = h θ i , and he nce L ( h θ i ) = h θ i . Next, suppo se tha t h φ i ⊆ h ξ i ⊆ GF( q ) ∗ h ξ i = h θ i . Then n = ord( φ ) | o r d( ξ ) and ord( ξ ) | ord( θ ) = nk , hence there ar e integers s, t with k = st suc h that ord( ξ ) = nk /s = nt . Moreo v er, since ( nt, q − 1) = ( n, q − 1)( dt, k ) = ( n, q − 1) t , we conclude from Theorem 4.4 that ξ has q -order ord q ( ξ ) = nt/ ( nt, q − 1) = d . So ξ and φ hav e the same q -order. Finally , since h φ i ⊆ h ξ i , w e ha v e that h ξ i = K h φ i with K the s ubgroup of G F( q ) ∗ , of order nt/d = et , where e = ( n, q − 1) (since nt = det | d ( q − 1 ), w e hav e et | ( q − 1) and suc h a subgroup K do es indeed exist); in fact, w e ha v e K = h ξ i ∩ GF( q ) ∗ . T o see this, first note that s ince φ d ∈ GF ( q ) ∗ has order e = n/d = ( n, q − 1) and K has order et , w e ha v e that h φ d i ⊆ K . Hence K h φ i = K h φ d i{ 1 , φ, . . . , φ d − 1 } = K { 1 , φ, . . . , φ d − 1 } , so that K = K h φ i ∩ GF( q ) ∗ and K h φ i has size | K | d = etd = nt = ord( ξ ), hence K h φ i = h ξ i . As a cons equenmce , if L is a q -p olynomial of q -degree m o v er GF( q m ) that fix es h φ i , then L ( h ξ i ) = L ( K h φ i ) = K L ( h φ i ) ⊆ K h φ i = h ξ i ; moreo v er since L is one-t o -one on GF( q ) ∗ { 1 , φ, . . . , φ d − 1 } = h θ i and h ξ i ⊆ h θ i , w e conclude that in fact L ( h ξ i ) = h ξ i . No w the desired conclusion follows from Theorem 2.5 . ✷ Corollary 4.6 If φ is nonstand ar d of de gr e e m over GF( q ) and if an element ξ in som e extension of GF( q ) has the same or der as φ , that is, if h ξ i = h φ i , then ξ is again non- standar d of de gr e e m over GF( q ) , with the same or der and q -or der as φ . Compare this “extension” results to T heorem 3.4 from [6]. Next, we presen t a tec hnique to “ lift” the nonstandardness of degree m o v er a subfield GF( q 0 ) of G F( q ) to nonstandardness ov er GF( q ), of the same order and sub-order, under certain conditions on q 0 and q . W e will refer to this Theorem as t he l i f ting the or em . Theorem 4.7 L et q 0 and q = q t 0 b e prime p owers, and let m b e a p ositive inte ger for which ( m, t ) = 1 . If ξ is nonstandar d of de gr e e m over F q 0 , then ξ also is no n standar d of de gr e e m over GF( q ) , of the same or der and w i th the q -or der of ξ e qual to its q 0 -or der. 13 Pro of: T o prov e the abov e claim, w e pro ceed as follo ws. First, w e note that F q m 0 ⊆ F q i with q = q t 0 holds precisely when m | ti , hence precisely when m | i . So ξ also has degree m o ve r GF( q ). Next, if ξ has degree m ov er GF( q 0 ), t hen ξ ∈ F q m 0 ; hence if ξ has order n , then n | q m 0 − 1. No w according to Theorem 4.4, the q 0 -order of ξ is d = n/ ( n, q 0 − 1) and its q -or der is n/ ( n, q − 1). It is w ell-known and easy to pro v e t ha t ( q m 0 − 1 , q t 0 − 1) = q ( m,t ) 0 − 1 = q 0 − 1 . Hence ( n, q − 1) = ( n, q t 0 − 1) = ( n, q m 0 − 1 , q t 0 − 1) = ( n, ( q m 0 − 1 , q t 0 − 1)) = ( n, q 0 − 1) , so that the q 0 -order and q - order o f ξ are equal. No w, since ( m, t ) = 1, there is an in teger u ≥ 1 suc h that ut ≡ 1 mo d m . W e claim that x q 0 = x q u (7) holds for all x ∈ F q m 0 . Indeed, since q = q t 0 , we hav e that (7) holds if and only if x q u /q 0 = x q tu − 1 0 = x for all x ∈ F q m 0 , which is t he c a se if and only if tu − 1 ≡ 0 mo d sm, whic h is how we hav e c ho osen u . Note also that ( u , m ) = 1, hence { 0 , u, 2 u, . . . , ( m − 1) u } = { 0 , 1 , 2 , . . . , m − 1 } mo d m. (8) No w ξ is no nstandard of degree m o v er F q 0 , so there is some nonstandard q 0 -p olynomial L ′ ( x ) = L 0 x + L 1 x q 0 + · · · + L m − 1 x q m − 1 0 of q -degree m in F q m 0 [ x ] f or whic h L ( h ξ i ) = h ξ i . Define the “lifted” q -p olynomial L ( x ) ∈ GF( q m )[ x ] b y L ( x ) = L 0 x + L 1 x q u + · · · + L m − 1 x q ( m − 1) u . According to (7), w e ha ve that L ′ ( x ) = L ( x ) on F q m 0 ; in part icular, w e hav e t ha t L ( h ξ i ) = L ′ ( h ξ i ) = h ξ i . Moreov er, ob viously L ( x ) is nonstandard if and o nly if L ′ ( x ) is nonstandard. ✷ Remark 4.8 Note that if ξ has the same minima l p olynomial over two fields K and L , then by definition ξ is nonstandar d over K if and only if ξ is nonstandar d over L . The pr o of of The or em 4.7 c an also b e interpr ete d as showing that under the c onditions of the the or em, the min imal p olynomi a l f ( x ) = m − 1 Y j = 0 ( x − ξ q j 0 ) = m − 1 Y j = 0 ( x − ξ q j ) of ξ over GF( q 0 ) and over GF( q ) ar e the same. 14 W e can no w use lifting and extension to construct nonstandard elemen ts of degree m o ve r GF( q ) with q -order d , where q = q t 0 with ( m, t ) = 1, from a nonstandard elemen t of degree m o v er GF( q 0 ) and q 0 -order also d , by applying Theorems 4.7 and 4 .5. W e will use this metho d to construct generalisations of Example 2. Example 5: Let q 0 = p s with p prime, let m ≥ 2 and q m 0 > 4. T ak e q = p r with r = st , and let ( t, m ) = 1. Finally , let ξ ∈ F q m 0 b e primitiv e, so ξ has order n = q m 0 − 1. In Example 2, w e ha ve shown tha t ξ is nonstandard of degree m ov er GF( q 0 ); its q 0 -order ob viously is ( q m 0 − 1) / ( q 0 − 1). So, acc o rding to Theorem 4.7, ξ is also nonstandard o v er GF( q ), of or der n = q m 0 − 1 and with q -order d = ( q m 0 − 1) / ( q 0 − 1). No w w e can use Theorem 4.5 to construct nonstandard elemen ts φ of q -o r der d = ( q m 0 − 1) / ( q 0 − 1) and o rder N = d ( q 0 − 1) k ov er GF( q ), for all k dividing ( q − 1) / ( q 0 − 1). All these elemen ts φ are p ow ers of an elemen t θ ∈ GF( q m ) of order d ( q − 1). These examples all hav e degree m ov er GF( q ), ha ve q -order d = ( q m 0 − 1) / ( q 0 − 1) > m and order n = de with q 0 − 1 | e | q − 1. Ob viously , eac h nonstandard elemen t ξ of degree m o ver GF( q ) with order n = de and q - o rder d = ( q m 0 − 1) / ( q 0 − 1) where q 0 − 1 | e and q m 0 > 4 can b e obt a ined in this w ay . Indeed, then φ = ξ e/ ( q 0 − 1) is primitive of degree m o ver F q 0 and nonstandard since q m 0 > 4, so ξ can b e obtained f r om the nonstandard φ b y lifting and extension. W e will refer to this class o f examples a s typ e II e xamples . ✷ 5 A subgroup in PGL( m, q ) re lated to a n onstandard elemen t o v e r GF( q ) In this section, w e will a ssume t ha t ξ ∈ G F( q m ) is of degree m ov er GF( q ), where q = p r for a prime p , with o rder o rd( ξ ) = n and q -order ord q ( ξ ) = d . So η = ξ d ∈ GF( q ) ∗ , and n = de , where e is the order of η . F urthermore, w e will assum e that ξ has minimal p olynomial f ( x ) = x m − σ m − 1 x m − 1 − · · · − σ 1 x − σ 0 o ve r GF( q ). Let the matrix T = T f =          0 0 0 · · · 0 0 σ 0 1 0 0 · · · 0 0 σ 1 0 1 0 · · · 0 0 σ 2 . . . . . . 0 0 0 · · · 0 1 σ m − 1          . denote the c omp anion matrix of f , the matrix represen tation of the map µ : a ( x ) 7→ xa ( x ) mo d f ( x ) o n GF( q )[ x ] mo d f ( x ) (m ultiplicatio n b y x mo dulo f ( x ) with resp ect to the basis 1 , x, . . . , x m − 1 . Equiv alen tly , T is t he matrix represen tation of m ultiplication b y ξ on GF( q m ) with resp ect to the basis 1 , ξ , . . . , ξ m − 1 of GF( q m ), considered as v ectorspace 15 o ve r GF( q ). Since f ( x ) | x d − η with η ∈ GF( q ) (or simply s ince ξ d = η ), we hav e that T d = η I . (9) W e now first restate and prov e The orem 4.3 from Section 2. Theorem 5.1 If f ( x ) ∈ GF( q ) is irr e ducible over GF( q ) a nd i f ξ is a zer o of f , then the r es tricte d p erio d δ ( f ) of f an d the q -or der o rd q ( ξ ) o f ξ satisfy δ ( f ) = ord q ( ξ ) . Pro of: W e first not e that a sequenc e u = { u k } k ≥ 0 is an f - sequenc e if and only if the v ectors u k ,m = ( u k , u k +1 , . . . , u k + m − 1 ) ⊤ satisfy u ⊤ k +1 ,m = u ⊤ k ,m T f for all k ≥ 0. As a conseq uence, if u ⊤ 0 ,m = ( u 0 , . . . , u m − 1 ) = (0 , . . . , 0 , 1 ), then u ⊤ 0 ,m T d = λu ⊤ 0 ,m holds if and only if u ⊤ i,m T d = u ⊤ 0 ,m T i + d = λu ⊤ 0 ,m T i = λu ⊤ i,m holds for i = 0 , . . . , m − 1. No w the matrix U with as its row s the v ectors u ⊤ i,m for i = 0 , . . . , m − 1 is triangular with nonzero an ti-dia g onal, hence in v ertible. So from the ab o v e, we conclude that u ⊤ 0 ,m T d = λu ⊤ 0 ,m if and only if U T d = λU if and o nly if T d = λI . ✷ F rom now on, we assume that, in addition, L ( x ) = L 0 x + L 1 x q + · · · + L m − 1 L m − 1 x q m − 1 of q - degree m in GF( q m )[ x ] that fixes h ξ i , that is, there exists a p erm utation π ∈ S n suc h that L ( ξ j ) = ξ π ( j ) for all j = 0 , . . . , n − 1 F o r later use, we will also assume that L (1) = 1. (As we remarke d earlier, this represen ts no loss of generality .) F or eac h i the standar d q -p olynomial L ( x ) = x q i has this prop ert y . Note that according to Theorem 2.5, ξ is nonstandard ov er GF( q ) if and only if there is a nonstandar d q -p olynomial as ab ov e. By abuse of notation, w e will also use L to denote the m × m matrix repres entation o ve r G F( q ) of the GF( q )- linear map x 7→ L ( x ) on G F( q m ) with resp ect to the basis 1 , ξ , . . . , ξ m − 1 . Note that if ξ j = m − 1 X i =0 c ( j ) i ξ i , with c ( j ) i in G F( q ) fo r i = 0 , . . . , m − 1 and all j ≥ 0, then ξ j is represen ted b y the v ector c ( j ) = ( c ( j ) 0 , . . . , c ( j ) m − 1 ) ⊤ 16 in G F( q ) m . As a consequence, the matrix L has as its columns the ve ctor s c ( π ( j )) for j = 0 , . . . , m − 1 . Let us write C to denote the collection of all v ectors c ( j ) . Then the ab ov e has the follo wing c onsequence . Theorem 5.2 We ha v e that T : c ( j ) 7→ c ( j +1) , L : c ( j ) 7→ c ( π ( j )) , so that the matrix gr oup G = h T , L i in GL( m, q ) fixes the c ol le ction C as a set. F or lat er use, we also consider the following “ normalisation”. W rite σ = σ m − 1 = T r GF( q m ) / GF( q ) , and assume that σ 6 = 0. Let ˜ ξ = ξ /σ . Then ˜ ξ ∈ G F( q m ) again has q -or der d and degree m ov er GF( q ), with minimal p olynomial ˜ f ( x ) = x m − ˜ σ m − 1 x m − 1 − · · · − ˜ σ 1 x − ˜ σ 0 , where ˜ σ i = σ i /σ m − i ; in particular, ˜ σ m − 1 = 1. So 1 , ˜ ξ , . . . , ˜ ξ m − 1 are another basis for GF( q m ) o v er GF( q ). W e no w write ˜ ξ j = m − 1 X i =0 ˜ c ( j ) i ˜ ξ i , with ˜ c ( j ) i in GF( q ) for i = 0 , . . . , m − 1 and a ll j ≥ 0, so that ˜ ξ j is represen ted b y the v ector ˜ c ( j ) = ( ˜ c ( j ) 0 , . . . , ˜ c ( j ) m − 1 ) ⊤ . Note that ˜ c ( j ) i = σ − j + i c ( j ) i for all j ≥ 0 a nd all i = 0 , . . . , m − 1. The c onjugate matrix M S of a matrix M b y an in ve rtible matrix S is defined a s M S = S M S − 1 . Note that the conjugate M S is the mat rix represen tation of the same linear map, but with resp ect to a basis tra nsfor ma t io n g iv en b y S . W e will write ˜ T to denote the companion matrix o f ˜ f ( x ). Define the diago nal mat r ix D as D = diag (1 , σ , . . . , σ m − 1 ) . Our observ at io ns are summarized in the following theorem. Theorem 5.3 With the a b ove definitions, we hav e that D c ( j ) = σ j ˜ c ( j ) . Mor e over, the c o njugate T D = D T D − 1 of T satisfies T D = σ ˜ T , and T D and the c onjugate L D = D LD − 1 of L satisfy T D : ˜ c ( j ) 7→ σ ˜ c ( j +1) , L D : ˜ c ( j ) 7→ σ π ( j ) − j ˜ c ( π ( j ) . So the c onjugate gr oup G D = h T D , L D i fixes the set ˜ C = { ˜ c ( j ) | j = 0 , . . . , n − 1 } as a set. 17 In the remainder o f this pa p er, w e will use the groups G and G D to o btain information on ξ and L , a nd, in particular, on the q -order o rd q ( ξ ) of the nonstandard elemen t ξ . T o this end, w e will consider the sets C and ˜ C as subsets of PG( m − 1 , q ), and t he groups G and G D as subgroups of PGL( m, q ), in its natural action on PG( m − 1 , q ). Here, PG( m − 1 , q ) consists of the lines through the origin in GF( q ) n . Equiv alen tly , PG( m − 1 , q ) consists of the nonzero ve cto r s v from GF( q ) n , where we iden tify a v ector v w ith its scalar multiples λv for λ ∈ GF( q ) ∗ . The group PGL( m, q ) consists of the collection G L( m, q ) of all nonsingular m × m matrices ov er GF( q ), where w e iden tify a matrix M with its scalar m ultiples λM , for λ ∈ GF( q ) ∗ . No w we assumed that ξ ha s q -order d , with ξ d = η ∈ GF( q ), so we hav e that the v ector c ( d ) represen ting ξ d satisfies c ( d ) = η c (0) . Since c ( j ) = T j c (0) for all j , w e see that the set C , considered as subset of PG( m − 1 , q ), has size d = ord q ( ξ ). No te furthermore that since T d = η I , the matrix T has order d as elemen t of the group PGL( m, q ). Note also that h ξ i = h η i ∪ h η i ξ ∪ . . . ∪ h η i ξ d − 1 , where the union is disjoint . As a consequence, there exists a p ermutation τ ∈ S d suc h that L ( ξ k ) = η k ξ τ ( k ) with η k ∈ h η i ⊆ G F( q ) ∗ , for all k = 0 , . . . , n − 1. So L , as an ele ment of PGL( m, q ), acts on C , considered as a subset of PG( m − 1 , q ) , by L : c ( j ) 7→ c ( τ ( j ) , for j = 0 , . . . , d − 1. W e summarize t he abov e in the next theorem. Theorem 5.4 The g r oups G and G D obtaine d fr om a nonstanda r d element ξ , c onsider e d as sub gr oups of PG( m, q ) , have orbits O = { c ( j ) | j = 0 , . . . , d − 1 } and O D = { ˜ c ( j ) | j = 0 , . . . , d − 1 } , r es p e ctively. Both O and O D have size d = ord q ( ξ ) and c ontain 1 = (1 , 0 , . . . , 0 ) ⊤ . 6 The case m = 2 W e no w in ve stiga t e the case where m = 2 in mo r e detail. So from now on, we will assume that m = 2 . So here ξ ∈ G F( q 2 ) \ GF( q ) is zero of the irreducible p o lynomial f ( x ) = x 2 − σ 1 x − σ 0 o ve r GF( q ), where w e a ssume that σ 1 6 = 0. (So w e assume that d = ord q ( ξ ) > 2.) W riting σ = σ 1 and λ = σ 0 /σ 2 1 , we also hav e that ˜ ξ = ξ /σ is zero of the p olynomial ˜ f = x 2 − x − λ . Note that, as a consequence , we hav e that ˜ ξ q = 1 − ˜ ξ . (10) 18 Again, w e assume that the q -p olynomial L ( x ) = L 0 x + L 1 x q of q -degree 2 ov er GF( q 2 ) fixes h ξ i as a set. A s remark ed b efore, w e may assume without loss of generalit y that L (1) = 1. Let ω , ν ∈ GF( q ) b e suc h that L (1) = 1 , L ( ξ ) = ω + ν ξ . Put ˜ ω = ω /σ . Then L (1) = 1 , L ( ˜ ξ ) = ˜ ω + ν ˜ ξ , so that the ma t r ix represen tations L and L D of the map induced b y the p olynomial L on GF( q 2 ) are giv en b y L = 1 ω 0 ν ! , L D = 1 ˜ ω 0 ν ! . Finally , it is easily v erified tha t the mat r ices T (m ultiplication by ξ ) a nd ˜ T = σ − 1 T D (m ultiplication b y ˜ ξ ) are giv en b y T = 0 σ 0 1 σ 1 ! , T D = σ 0 λ 1 1 ! . In what follows, w e will in ve stiga te the subgroup Ξ = h Λ , Γ i of PGL(2 , q ) generated by the elemen ts Λ = 0 λ 1 1 ! , Γ = 1 ˜ ω 0 ν ! . (11) W e will emplo y the usual identification of PG(1 , q ) with the set GF( q ) ∪ {∞} b y iden tifying the elemen t ( x, y ) ∈ PG(1 , q ) with the finite field elemen t x/y ∈ GF( q ) if y 6 = 0 and with ∞ if y = 0 . As a consequence, a matrix M = a b c d ! from PGL(2 , q ) now acts on a n elemen t x from GF( q ) + = GF( q ) ∪ {∞} as M : x 7→ ( ax + b ) / ( cx + d ) . So now the field eleme nt ξ j ∈ GF( q 2 ) corresp onds t o c ( j ) = ( c ( j ) 0 , c ( j ) 1 ) ∼ c ( j ) 0 /c ( j ) 1 in GF( q ) + ; in particular, we hav e that 1 = c (0) ∼ ∞ and ξ = c (1) ∼ 0. In the next theorem, w e summarize the main consequences of the a b o v e definitions and assumptions. Theorem 6.1 L et ξ ha ve de gr e e m over GF( q ) , with q -or der d = ord q ( ξ ) > 2 and minimal p olynomial f ( x ) = x 2 − σ x − λσ 2 . L et L b e a q -p olynomial o f q -de g r e e m ove r GF( q m ) that fixes h ξ i , with L (1 ) = 1 and L ( ξ ) = σ ˜ ω + ν ξ , and let Λ and Γ b e the asso ciate d matric es as in (11). Then the fol lowing holds. 19 (i) The element Λ has or der d in PGL(2 , q ) , and no fixe d p oints on GF( q ) + . Mor e over, we ha ve that Λ k = λF k − 1 λF k F k F k +1 ! , wher e the F k ar e define d by F 0 = 0 , F 1 = 1 , and F k +2 = F k +1 + λF k for al l k . I n p articular, Λ has or der d and F k = 0 if and only if k ≡ 0 mo d d . The matrix Λ in duc es a map x 7→ λ/ (1 + x ) on GF( q ) + . (ii) T he element Γ induc es a map x 7→ ( x + ˜ ω ) /ν on GF( q ) + . We have that Γ k = 1 ˜ ω (1 + ν + · · · + ν k − 1 ) 0 ν k ! . In p articular, if ν has or der e , then Γ has or der e (i f ν 6 = 1 or ˜ ω = 0 ) or p (if ν = 1 a n d ˜ ω 6 = 0 ). (iii) The s ubse t O = {∞ , Λ( ∞ ) , . . . , Λ d − 1 ( ∞ ) } of GF( q ) + is an orbit of the sub gr o up Ξ of PGL(2 , q ) gen e r ate d by the maps Λ and Γ . The “standar d” q -p ol ynom ials L ( x ) = x or L ( x ) = x q c o rr esp ond to the c ases ν = 1 , ω = 0 , ˜ ω = 0 , and ν = − 1 , ω = σ 1 , ˜ ω = 1 , r es p e ctively. Pro of: Most of the claims are a direct consequences of our a ssumptions and definitions. Hence all or bits of Λ on GF( q ) + ha ve the same size d . The claim concerning the case where L ( x ) = x is eviden t. Finally , since ξ and ξ q are the zero es of f ( x ) = x 2 − σ 1 x − σ 0 , w e hav e that σ 1 = ξ + ξ q , σ 0 = − ξ q +1 . Hence if L ( x ) = x q , then ν ξ + ω = L ( ξ ) = ξ q = σ 1 − ξ , so that ν = − 1 and ω = σ 1 . ✷ It turns out that the cases d = 3 , 4 , 5 need a sp ecial treatment. F or la ter use, w e now collect the required extra infor mation. Note that according to Theorem 6.1 , the orbit O has size d and is giv en by O = {∞ , 0 , λ, λ/ ( 1 + λ ) , λ ( 1 + λ ) / (1 + 2 λ ) , λ ( 1 + 2 λ ) / (1 + 3 λ + λ 2 ) , . . . } . (12) Lemma 6.2 Ther e ar e no no n standar d ξ of de gr e e 2 o v er GF( q ) with q -or der d = 3 . Pro of: F rom (1 2) we see that if d = 3 , t hen necessarily λ = − 1. No w since O is also in v arian t unde r Γ, we hav e that O = {∞ , 0 , − 1 } = {∞ , ˜ ω /ν, ( − 1 + ˜ ω ) / ν } . So we hav e one of t wo cases: 1. ˜ ω = 0. Then ν = 1, so w e are in the case where L ( x ) = x . 20 2. ˜ ω = 1. Then ν = − 1, so w e are in t he case where L ( x ) = x q . Since there are no other p ossibilities, the claim follows. ✷ Lemma 6.3 If ξ is nonstandar d of de gr e e 2 over GF( q ) with q -or der d = 4 , then p = 3 and ˜ ξ = ξ /σ is p rimitive in GF(9) . Mor e over, Ξ is actual ly a sub gr oup of PG L (2 , 3) . Pro of: F rom ( 1 2) w e see tha t if d = 4, then necessarily λ = − 1 / 2. Now since O is also in v arian t unde r Γ, we hav e that O = {∞ , 0 , − 1 / 2 , − 1 } = {∞ , ˜ ω /ν, ( − 1 / 2 + ˜ ω ) /ν , ( − 1 + ˜ ω ) /ν } . So we hav e one of three cases. 1. ˜ ω = 0. Then {− 1 / 2 , − 1 } = { ( − 1 / 2) /ν , − 1 / ν } , so either ν = 1 (w hich corresp onds to the case where L ( x ) = x ), o r ν = 1 / 2 = 2, so the characteristic p = 3 a nd ν = − 1. 2. ˜ ω = 1 / 2. Then {− 1 / 2 , − 1 } = { (1 / 2 ) /ν, ( − 1 / 2) /nu } , so p = 3, ˜ ω = 1 / 2 = − 1, and either ν = − 1 or ν = 1 . 3. ˜ ω = 1. Then {− 1 / 2 , − 1 } = { 1 /ν, (1 / 2) /ν } , so either ν = − 1 (whic h corresp onds to the case where L ( x ) = x q ), or ν = − 1 / 2 = − 2, so tha t p = 3, ν = 1, and ˜ ω = 1. W e are left with four cases. All hav e p = 3, so that ˜ ξ is zero of x 2 − x − λ = x 2 − x − 1 and ˜ ξ is primitiv e in GF(9). Note tha t thes e four remaining cases represen t the differen t nonstandard w ays of mapping the no nstandard subgroup GF(9) ∗ on to itse lf. In all these cases, λ , ˜ ω , and ν are in GF(3), hence Ξ is actually a subgroup of PG L(2 , 3). ✷ Lemma 6.4 If ξ is nonstandar d of de gr e e 2 over GF( q ) with q -or d e r d = 5 , then p = 2 , and ˜ ξ = ξ /σ is p rimitive in GF(16) . Mor e over, Ξ is actual ly a sub gr oup of PGL(2 , 4) . Pro of: F rom (12) we see that if d = 5, then necessarily λ 2 + 3 λ + 1 = 0, so that λ = − ( λ + 1) 2 . F or later use, w e remark that in c haracteristic p = 2 , w e ha ve that λ is primitiv e in G F(4) and ˜ ξ , the zero of x 2 + x + λ , is primitiv e in GF(16). So we ar e done if w e can pro ve that in all cases p = 2, or ( ν, ˜ ω ) = (1 , 0) (corresp onding to the case where L ( x ) = x ), or ( ν , ˜ ω ) = ( − 1 , 1) (corresp onding to the case where L ( x ) = x q ). No w s ince O is also inv arian t under Γ, w e ha v e that O = {∞ , 0 , λ , − 1 − λ, − 1 } = {∞ , ˜ ω/ν , ( λ + ˜ ω ) /ν, ( − 1 − λ + ˜ ω ) /ν, ( − 1 + ˜ ω ) / ν } . So w e hav e one of four cases for ˜ ω . 1. ˜ ω = 0 and { λ, − 1 − λ, − 1 } = { λ/ν , ( − 1 − λ ) /ν , − 1 / ν } . T hen ν ∈ { 1 , λ/ ( − (1 + λ )) , − λ } = { 1 , 1 + λ, − λ } , so w e ha v e one of the following. 21 (a) ν = 1 (corresp onding to the case w here L ( x ) = x ); (b) ν = 1 + λ , { λ, − 1 } = {− 1 , − 1 / ( 1 + λ ) } . Hence λ = − 1 / (1 + λ ), or λ 2 + λ + 1 = 0; com bined with the o ther equation for λ this sho ws that, in addition, p = 2. (c) ν = − λ , { λ, − 1 − λ } = { (1 + λ ) /λ, 1 /λ } . So either λ = (1 + λ ) /λ and − 1 − λ = 1 /λ , whence p = 2, or λ = 1 / λ , which leads to an imp ossibilit y . 2. ˜ ω = − λ a nd { λ, − 1 − λ, − 1 } = {− λ/ν , ( − 2 λ − 1) /ν, ( − λ − 1) /ν } = {− λ/ν , ( λ 2 + λ ) /ν, ( − λ − 1) /ν } . Then ν ∈ {− 1 , λ/ (1 + λ ) = − (1 + λ ) , λ } , so w e ha v e o ne of the follo wing. (a) ν = − 1. This leads t o p = 2 or an imp ossibilit y . (b) ν = − (1 + λ ). Then either p = 2, or an imp ossibilit y . (c) ν = λ . Here either p = 2 o r a n imp ossibility . 3. ˜ ω = 1 + λ a nd { λ, − 1 − λ, − 1 } = { (1 + λ ) /nu, (2 λ + 1) /ν, λ/ν } = { (1 + λ ) /ν , − ( λ 2 + λ ) /ν, λ/ν } , so w e hav e one of the following. (a) ν = − (1 + λ ). Then we hav e − (1 + λ ) = − λ/ (1 + λ ), whic h leads to p = 2. (b) ν = − ( λ 2 + λ ). Then either an imp ossiblit y or λ = − 1 / (1 + λ ), whic h leads to p = 2. (c) ν = − λ . Then either p = 2, or an imp ossibilit y . 4. ˜ ω = 1 a nd { λ, − 1 − λ, − 1 } = { 1 /ν, (1 + λ ) /ν , − λ/ν } , so w e ha v e one o f the following. (a) ν = − 1. This correspo nds to the case where L ( x ) = x q . (b) ν = − (1 + λ ). This leads to p = 2 or an imp ossibilit y . (c) ν = λ . Here either λ = 1 / λ and − 1 − λ = ( 1 + λ ) /λ , which leads to p = 2, o r λ = (1 + λ ) / λ and − 1 − λ = 1 /λ , whic h again leads to p = 2. So in all nonstandard cases w e ha v e p = 2. Moreov er, in all these cases, λ , ˜ ω , and ν are in GF(4), hence Ξ is actually a subgroup of PGL(2 , 4). ✷ 7 A sub g roup in PGL(2 , q ) The groups PGL(2 , q ) are one of the few groups for whic h the complete subgroup structure is kno wn. In this section, w e will use this kno wledge to obtain further information on the subgroup Ξ of PGL(2 , q ) from Theorem 6.1 . F or our purp oses, t he following is sufficien t. 22 Theorem 7.1 ([13], [16], [11]) L et q = p r with p prime. (i) I f M is a non-identity eleme nt in PGL(2 , q ) of or der k , with f fixe d p oints, then al l orbits of size > 1 have size k , and either f = 1 , k = p , or f = 2 , k | q − 1 , or f = 0 , k | q + 1 . (ii) T he sub gr oups of PGL(2 , q ) ar e as fol lo ws: 1. C yclic sub gr oups C k , of or der k = 2 (if p is o dd), or of or de r k > 2 with k | q ± 1 . 2. D ihe dr al sub g r oups D 2 k of or der 2 k , with k = 2 (if p is o dd ), or with k > 2 and k | q ± 1 . 3. Ele mentary ab elian sub gr oups E p k , of or der p k with 0 ≤ k ≤ r . 4. A semidir e ct pr o duct of the elem entary sub gr oup E p k , wher e 1 ≤ k ≤ r , and the cyclic gr oup C ℓ , wher e ℓ | q − 1 and ℓ | p k − 1 . 5. S ub gr oups isomorphic to A 4 ∼ = PSL(2 , 3) , S 4 ∼ = PGL(2 , 3) , or A 5 ∼ = PSL(2 , 4) . 6. O ne c onjugacy class o f sub gr oups isomorphic to PSL(2 , p k ) , wh e r e k | r . 7. O ne c onjugacy class o f sub gr oups isomorphic to PGL(2 , p k ) , wher e k | r . In the references, the classifications a re given for subgroups of PSL(2 , q ). If q is eve n, then PSL(2 , q ) = PGL(2 , q ). T o obtain the classification for PGL(2 , q ), note that if q is o dd, then PGL(2 , q ) is a subgroup of PSL(2 , q 2 ) and has a unique subgroup PSL(2 , q ), of index t wo. A similar classification ha s b een used e.g. in [8] and [9] in the case where q is o dd. W e now use this classification to show the following. Theorem 7.2 The gr oup Ξ fr om The or em 6.1 is one of the fol lowing. • A cyclic gr oup, in the c ase wher e L ( x ) = x ; • a dihe dr al gr oup, in the c ase wher e L ( x ) = x q ; • a gr oup of the form PSL (2 , q 0 ) or PGL(2 , q 0 ) , in the no nstandar d c ase, with d = q 0 + 1 > 3 and q = q t 0 , wher e t is o dd. Pro of: W e break the pro of into a num b er of cases. (1) The subgroup Ξ c a nnot be cyclic except when L ( x ) = x . Indeed, if Ξ is c yclic, then Λ and Γ comm ute, that is, ΛΓ = ΓΛ. It is easily v erified that this happ ens if and only if ˜ ω = 0 and ν = 1, t hat is, if Γ = I . (2) The subgro up Ξ cannot b e dihedral except when L ( x ) = x q . Indeed, Λ has o rder d > 2, so if Ξ is dihedral, then b oth Γ and ΓΛ ha ve order t w o . Hence ν = − 1 and ˜ ω = − ν , so a ccording to Theorem 6.1, w e ha v e L ( x ) = x q . (3) The subgroup Ξ cannot be elem entary ab elian of order p k . Indeed, since d | q + 1 , w e ha ve ( d , p ) = 1, so the order of Λ cannot b e a p ow er of p . 23 (4) The subgroup Ξ cannot b e semisimple pro duct of an elemen tary ab elian group of order p k with a cyclic g r o up of order ℓ , if ℓ | q − 1 and ℓ | p k − 1. Indeed, supp ose that this w ould b e the case. The semidirec t pro duct has cardinalit y p k ℓ , and since ( d, p ) = 1, w e w ould conclude that d | ℓ , hence d | q − 1. Now w e also ha ve that d | q + 1, so it w ould follo w that d | 2, whic h is imp ossible if d > 2. (5) If the subgroup Ξ is one of A 4 , S 4 , or A 5 , then the order d > 2 of the elemen t Λ ∈ Ξ is o ne of 3, 4, o r 5. Thes e cases w ere handled in Section 6. In Lemma 6.2, it was sho wn that the case d = 3 is not p ossible. In Lemma 6.3 it w as sho wn that if d = 4, t hen p = 3 and the group Ξ is a subgroup of PGL(2 , q 0 ) for q 0 = 3. Finally , in Lemma 6 .4 , it w as sho wn that if d = 5, then p = 2 and the g r o up Ξ is a subgroup of PGL(2 , q 0 ) with q 0 = 4. As a conseque nce, since w e are not in one of the cases (1-4) ab ov e, w e m ust ha v e one of the cases (6), (7) b elo w. (6), (7) Here we hav e that Ξ is isomorphic to either PSL(2 , q 0 ) or PGL(2 , q 0 ), with GF( q 0 ) a subfield of GF( q ). Suc h a subgroup is conjugated in PGL(2 , q ) t o the “ob vious” subgroup consisting o f in vertible matrices with en tries in GF( q 0 ). It is easily v erified that these tw o groups b oth ha v e one orbit GF( q 0 ) + of size q 0 + 1, and one orbit GF( q 2 0 ) \ GF( q 0 ) of size q 2 0 − q 0 . Moreov er, it is easy to sho w that b oth groups act regularly on G F( q 2 ) \ G F( q 2 0 ). Indeed, this immediately follo ws from the fact that the fixed points in GF ( q ) + of a non- iden tity map M : x 7→ ( ax + b ) / ( cx + d ) with a, b, c, d ∈ GF( q 0 ) are the zeros of the non-trivial p olynomial cx 2 + ( d − a ) x − b of degree tw o ov er GF( q 0 ), hence are con tained in GF( q 2 0 ). (F or more details, see, e.g., [12].) So all other orbits of Ξ are of size | Ξ | , hence ha ve a size equal to q 0 ( q 2 0 − 1) or q 0 ( q 2 0 − 1) / 2 (if q is odd and G q = PSL(2 , q 0 ). Since Ξ has an o r bit of size d and since d | q + 1, w e ha v e ( d, q ) = 1 and hence w e m ust hav e t ha t d = q 0 + 1. No w note that since Λ has order d a nd no fixed p oints, all orbits of Λ hav e size d , hence d | q + 1. So in t he no nstandard case, w e hav e q = p r and d = q 0 + 1 , with q 0 of the form q 0 = p s and with s | r , that is, with q = q t 0 for some t . Since d | q + 1, it follow s that t is o dd. ✷ Next, w e w an t to show that if G is isomorphic to PSL(2 , q 0 ) or PGL(2 , q 0 ), then λ , ν , and ˜ ω are actually contained in GF( q 0 ). T o this end, w e need some preparatio n. If M = ( M i,j ) is a matrix ov er GF( q ), where q = p r , then we write M ( p s ) to denote the matrix with en tries M p s i,j . Lemma 7.3 ([12]) L et q = q t 0 and let M ∈ PGL(2 , q ) . Then M is c on tain e d in PGL(2 , q 0 ) if and only if M ( q 0 ) = φM holds for some φ ∈ GF( q ) ∗ . Pro of: The pro of in [12] uses Galois theory . F or completeness’ sake , w e sk etch a simple pro of here. (In f a ct, many differen t pro ofs a re p o ssible.) The matrix M = a b c d ! is in PGL(2 , q 0 ) prec isely when some m ultiple β M of M has all its en tries in G F( q 0 ), so when the map x 7→ ( ax + b ) / ( cx + d ) fixes G F( q 0 ) + as a set. Now (( ax + b ) / ( c x + d ) ) q 0 = 24 ( ax + b ) / ( c x + d ) for all x ∈ G F( q 0 ) + leads to a second degree equation that is iden tically zero on GF( q 0 ), so has a ll co efficien ts equal to zero. F rom the resulting three equations, the lemma follow s. ✷ Next, f or a matrix M o v er GF( q ), w e write det( M ) and T r( M ) to denote the determinan t and trace of M , respective ly . Also, w e write M A to denote the conjugate AM A − 1 of M b y A . ov er Lemma 7.4 L et GF( q 0 ) b e a subfield of GF( q ) . (i) A matrix M o v e r GF( q ) is c ontaine d in a sub gr oup of PGL(2 , q ) iso m orphic to PSL(2 , q 0 ) or PGL(2 , q 0 ) if and only if ( M A ) ( q 0 ) = φM A for som e matrix A in PGL(2 , q ) and s ome φ ∈ GF( q ) ∗ , wher e φ 2 = det( M ) q 0 − 1 and e ither φ = T r( M ) q 0 − 1 or T r( M ) = 0 . (ii) If a matrix M over GF( q ) is c ontaine d in a sub gr oup o f PGL(2 , q ) i s o morphic t o PSL(2 , q 0 ) or PGL(2 , q 0 ) , then either T r( M ) = 0 or T r( M ) 2( q 0 − 1) = det( M ) q 0 − 1 . Pro of: If M is con tained in some subgroup of PGL(2 , q ) isomorphic to PSL(2 , q 0 ) or PGL(2 , q 0 ), then there is a matrix A ∈ PGL(2 , q ) suc h that M A is con tained in PSL(2 , q 0 ) or PGL(2 , q 0 ), that is, according to L emma 7 .3, ( M A ) ( q 0 ) = φM A , (13) for some A ∈ PGL(2 , q ) and some φ ∈ GF( q ) ∗ . Now det( M A ) = det( A ) and T r( X A ) = T r( X ), so fr o m (13), w e conclude that det( M ) q 0 = φ 2 det( M ) , T r( M ) q 0 = φ T r( M ) , hence φ 2 = det( M ) q 0 − 1 and either T r( M ) = 0 or φ = T r( M ) q 0 − 1 . ✷ No w w e a pply t his result to our matrices Λ and Γ. The result is as follows . Theorem 7.5 In the no n standar d c ase, ther e exists a prim e p ower q 0 such that d = q 0 + 1 > 3 , q = q t 0 with t o dd, a n d the sub gr o up Ξ = h Λ , Γ i of PGL(2 , q ) gener ate d by Λ and Γ as in The or em 6. 1 is e qual to either P S L (2 , q 0 ) or PGL(2 , q 0 ) . Mor e over, we have λ, ν, ˜ ω ∈ GF( q 0 ) , and GF( q 0 ) is the smal les t subfield of GF( q ) c ontaini n g λ . Pro of: Acccording to The o rem 7.2, in the nonstandard case we ha v e q = q t 0 with t o dd, d = q 0 + 1, and Ξ conjugate in PGL(2 , q ) to either PSL(2 , q 0 ) or P GL(2 , q 0 ). Now firs t , since det(Λ) = − λ and T r(Λ) = 1, w e se e from Lemma 7.4 tha t λ m ust b e con tained in GF( q 0 ). Next, sinc e the orbit h Λ i ( ∞ ) o f Λ con taining ∞ has s ize d = q 0 + 1, it m ust be equal to GF( q 0 ) + ; since it is fixed b y Ξ, w e m ust n o w hav e Γ(GF( q 0 ) + ) = GF( q 0 ) + . This immediately implies that b o th ν and ˜ ω m ust b e con tained in G F( q 0 ). ✷ W ev will no w use this result to sho w the following. Theorem 7.6 A nonstandar d eleme n ts of de gr e e two over a fi e ld GF ( q ) with q -or der d is either of typ e II, with d = 2 and of the form as in Example 1, so h a s n = 2 e with b oth q and ( q − 1) /e o dd, or is of typ e I and has d ≥ 4 of the form d = q 0 + 1 , for some q 0 such that q = q t 0 with t o dd, and c an b e obtaine d fr om a nonstandar d element of de gr e e two o ver G F( q 0 ) with q 0 -or der q 0 + 1 by li f ting and extension as in The or ems 4.7 and 4.5. 25 Pro of: Let ξ be nonstandard of degree m = 2 ov er GF( q ), with minimal p o lynomial f ( x ) = x 2 − σ x − σ 2 λ o v er GF( q ), and let ξ has order n = de and q -order d = ord q ( ξ ). According to Theorem 4.4, we hav e e = ( n, q − 1) | q − 1 and ( d , ( q − 1) /e ) = 1. No w d = 2 if and only if σ = 0; in that case ξ is of type I I, so as in Example 1. So in addition w e will assume tha t d > 2 and σ 6 = 0. W rite ˜ ξ = ξ /σ . Then ˜ ξ has minimal p olynomial ˜ f = x 2 − x − λ . Let L ( x ) b e a nonstandard q - p olynomial of q -degree t w o o ve r GF( q 2 ) that fixes h ξ i , with L (1) = 1 and L ( ξ ) = ω + ν ξ . W rite ˜ ω = ω /σ . Then L ( ˜ ξ ) = ˜ ω + nu ˜ ξ . According to Theorem 7.5, we no w hav e tha t d = q 0 + 1 ≥ 4, where q = q t 0 with t o dd, and λ, ˜ ω , ν ∈ GF ( q 0 ), with ( ν , ˜ ω ) 6 = (1 , 0) , ( − 1 , 1). W e claim that L is a G F( q 0 )-linear map of q 0 -degree t w o on GF( q 2 0 ). This can be s ho wn as in the pro of o f the “lifting” t heorem, but can also sho wn directly , as follo ws. Since ˜ ξ has minimal p olynomial ˜ f ( x ) = x 2 − x − λ with λ ∈ G F( q 0 ), w e ha v e that 1 , ˜ ξ is a basis for GF( q 2 0 ) ov er GF( q 0 ); moreov er, since t is o dd, w e hav e ˜ ξ q = ˜ ξ q t 0 = ˜ ξ q 0 . Hence L is q 0 -linear ov er G F( q 2 0 ). No w L is a bijection on h ξ i and maps G F( q 2 0 ) into GF( q 2 0 ); w e conclude that L is also a bijection on h ξ i ∩ GF( q 2 0 ) = h φ i , where φ = ξ δ 0 with δ 0 the q 2 0 -order of ξ . Hence φ is nonstandard of degree 2, b oth ov er GF( q ) and o ver GF( q 0 ). F or later use, w e w ant to sho w that ( q − 1 ) /e mus t b e o dd. Indeed, w e ha v e n = de with d = q 0 + 1 and e | q − 1. No w, as easily v erified, ( q 0 + 1 , q − 1) = ( q 0 + 1 , q t 0 − 1) = ( 1 , if q 0 is ev en ; 2 , if q 0 is o dd , hence q 0 + 1 = d = n/ ( n, q − 1) = ( q 0 + 1) e/ (( q 0 + 1) e, q − 1) = ( q 0 + 1) / ( q 0 + 1 , ( q − 1) /e ) holds precisely when ( q − 1) /e is o dd. Next, b y Theorem 4.4, w e ha v e that the q 2 0 -order δ 0 of ξ is giv en b y δ 0 = n/ ( n, q 2 0 − 1) = e/ ( e, q 0 − 1) = e/e 0 , where e 0 = ( e, q 0 − 1). No te that φ = ξ δ 0 has order n 0 = n/δ 0 = ( q 0 + 1 ) e 0 . W e claim that the q 0 -order d 0 of φ is equal to d . Indeed, d 0 = n 0 / ( n 0 , q 0 − 1) = ( q 0 + 1) e 0 / (( q 0 + 1) e 0 , q 0 − 1) = ( q 0 + 1) / ( q 0 + 1 , ( q 0 − 1) /e 0 , so w e are done if ( q 0 − 1) /e 0 = ( q 0 − 1) / ( e, q 0 − 1) is o dd, whic h follows immediately from the fa ct that q 0 − 1 | q − 1 (so e contains ev ery fa ctor 2 con ta ined in q − 1, so certainly all factors 2 con tained in q 0 − 1). Finally , we w ant to sho w that ξ can b e obtained fro m φ b y lifting and extension. No w lifting show s that, as remark ed earlier, φ is also nonstandard of degree 2 ov er GF( q ). W e kno w b y definition of φ that h φ i ⊆ h ξ i , hence according to Theorem 4.5 w e only hav e to sho w that ξ ∈ GF( q ) ∗ h φ i . T o this end, write η = ξ q 0 +1 . Then η ∈ GF( q ) ∗ , so it is 26 sufficien t to sho w that ξ ∈ h η ih φ i . Since η = ξ q 0 +1 and φ = ξ e/e 0 , this su bgroup contains all p ow ers ξ k of ξ where k is of the form k = i ( q 0 + 1) + j e/e 0 . So we a r e done if w e can show that ( q 0 + 1 , e/e 0 ) = 1; since e | q − 1 and q = q t 0 with t o dd, w e hav e ( q 0 + 1 , q − 1) | 2 and w e hav e to sho w that e/e 0 is o dd. This is eviden t in the case where q is even , s o w e also ass ume that q is o dd. W rite q − 1 = 2 r s, q 0 − 1 = 2 r 0 s 0 . No w q − 1 = q t 0 − 1 = ( q 0 − 1)(1 + q 0 + · · · + q t − 1 0 ) , and q 0 ≡ 1 mo d 2, he nce 1 + q 0 + · · · + q t − 1 0 ≡ t ≡ 1 mo d 2; w e conclude that r 0 = r . Moreov er, ( q − 1 ) /e is o dd, so e = 2 r f with f o dd; therefore e 0 = ( e, q − 1 ) is also divisible by 2 r and hence e/e 0 is indeed o dd. ✷ No w in Theorem 2 .4 of [7], it is sho wn that a nonstandard finite field elemen t of degree t wo ov er GF( q ) with q -order q + 1 is necessarily p rimitiv e, that is, has or der q 2 − 1. Corollary 7.7 A n o nstandar d eleme nt o f de gr e e two ove r a fi e ld GF( q ) either is of typ e I, with q -or der 2 as in Exam p le 1, or is o f typ e II , with q -or der of the form q 0 + 1 ≥ 4 with q = q t 0 for an o dd inte ger t , as in Example 5. Ac kno w l edgemen t W e gratefully ac knowle dge email con v ersations with Brison and Nogueira in whic h they p oin ted out that the nonexistance of nonstandard elemen ts of degree tw o of other t yp es follo ws fro m our results in com bination w ith some of their unpublished material. References [1] Thierry P . Berger and P a scale Charpin, The p e rm utation gr oup of affine-invariant extende d cyclic c o des , IEEE T rans. on Inform. Theory , v ol. 42, no. 6, Nov em b er 199 6 , pp. 2194–22 09. [2] Thierry P . Berger, The automorphism gr o up of double-err or-c o rr e cting BCH c o des , IEEE T rans. on Inform. Theory , vol. 40, no. 2 , March 1994, pp. 5 38–542. [3] O w en J. Brison a nd J. 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