Subclose Families, Threshold Graphs, and the Weight Hierarchy of Grassmann and Schubert Codes

SUBCLOSE F AMILIES, THRESHO LD GRAPHS, AND THE WEIGHT HIERAR CHY OF GRASSMANN AND SCHUBER T CODES SUDHIR R. GHORP ADE, ARUNKUMAR R. P A TIL, AND HARISH K. PILLAI De dic ate d to Gil les La chaud on his sixtieth birthda y Abstract. W e discuss the problem of determining the complete weigh t hier- arc h y of linear error correcting codes asso ciated to Grassmann v arieties and, more gene rally , to Sc hubert v arieties i n Grassmannians. In g eometric terms, this corresp onds to determini ng the maximum n umber of F q -rational p oints on sections of Sch ubert v ari eties ( with nondegenerate Pl ¨ uc k er embedding) by linear sub v arieties of a fixed (co )dimension. The problem is pa rtially solve d in the case of Grassmann co des, and one of the solutions uses the combinatorial notion of a closed f amily . W e prop ose a generalization of this to what i s called a subclose f ami ly . A nu mber of pr operties of sub close famil ies are prov ed, and its connection with the notion of threshold graphs and graphs with maximum sum of squares of ve rtex degrees i s outlined. 1. Introduction It has bee n almost a decade since the first named author and Gilles Lachaud wrote [5] where alter native proofs of Nogin’s results on higher weigh ts of Grass ma nn co des [14] were given and Sc h ubert c o des were in troduce d. Or iginally , muc h of [5] was conceived a s a side remar k in [6]. But in retrosp ect, it appear s to have b een a go o d idea to write [5] as an indep endent article and use the opp or tunit y to prop ose therein a conjecture concerning the minimum distance o f Sch ubert codes. This conjecture has been of some interest, and after b eing proved, in the affirmative, in a num ber of special cases (cf. [1 , 17, 7 , 9]), the general case app ear s to hav e been settled very rece nt ly by Xiang [19 ]. The time sees rip e, therefor e , to up the ante and think ab o ut more g eneral questions. It is wit h this in view, that we discuss in this pap er the problem of determining the complete weigh t hier arch y o f Sch ubert codes and, in particular, the Gra ssmann co des. In fact, the case of Grassmann co des and the determination of higher weights in the ca ses not cov ered by the result in [14] and [5] has a lready b een considered in some rec ent work (cf. [10, 11, 8 ]). What is prop osed here is basically a plausible approach to tac kle the general c a se. This pa pe r is orga nized as follows. In Section 2 below we r ecall the combinatorial notion of a clo se family , a nd in tro duce a mo r e general notion of a sub clo se family . A num ber of elementary prop erties o f sub close families are proved here, including a nice dualit y that prev ails amo ng these. Basic notions co ncerning linear error correcting co des , such as the minim um distance and more gener ally , the higher weigh ts are r eviewed in Section 3 . F ur ther, w e state here a genera l conjecture that relates the higher weigh ts of Gr assmann co des and Sch ubert codes with sub close families. Finally , in Sectio n 4 , we reca ll threshold graphs a nd optimal graphs and Date : September 7, 2021. 2000 Mathematics Subje ct Classific ation. 94B05, 05C07, 05C35, 14M15. Key wor ds and phr ases. Linear code, higher weigh t, Grassmann v ari et y , Grassmann co de, Sc h ubert v ariet y , Sch ub ert co de, threshold graph, optimal graph. 1 2 SUDHIR R. GHORP ADE, AR UNKUMAR R. P A TIL, AND HARIS H K. PILLAI then show that, in a sp ecia l case, subclose f amilies are clo sely related to these w ell- studied notio ns in gra ph theory . As an application we obtain explicit bo unds on the sum of squares of degrees of a s imple graph in terms of the n umber of vertices and edg es, which seem to amelior ate and co mplement some of the known results on this topic that has been of some in terest in graph theory (cf . [3 , 2, 15]). 2. Cl ose F amilies and Subclo se F amilies Fix in tegers ℓ, m suc h t hat 1 ≤ ℓ ≤ m . Set k :=  m ℓ  and µ := max { ℓ , m − ℓ } + 1 . Let [ m ] denote the set { 1 , . . . , m } of first m p ositive integers. Giv en a ny nonnega tive int eger j , let I j [ m ] denote the set o f all subsets of [ m ] of car dinality j . Let Λ ⊆ I ℓ [ m ]. F ollowing [6], we call Λ a close family if | A ∩ B | = ℓ − 1 for all A, B ∈ Λ. Supp ose | Λ | = r . Then Λ is said to be of T yp e I if if there ex ists S ∈ I ℓ − 1 [ m ] and T ⊆ [ m ] \ S with | T | = r such t hat Λ = { S ∪ { t } : t ∈ T } , whereas Λ is said to b e o f T yp e II if there exis ts S ∈ I ℓ − r +1 [ m ] and T ⊆ [ m ] \ S with | T | = r such that Λ = { S ∪ T \ { t } : t ∈ T } . Basic results ab o ut close families are a s follows. Prop ositi on 1 (Structur e Theorem for Close F amilies) . L et Λ ⊆ I ℓ [ m ] . Then Λ is close if and only if Λ is either of T yp e I or of T yp e II. This is prov ed in [6, Thm. 4.2]. An immediate consequence is the follo wing. Corollary 2. Le t r b e a n onne gative inte ger. A close famil y in I ℓ [ m ] of c ar dinal ity r exists if and only if r ≤ µ . I n gr e ater detai ls, a close family of T yp e I in I ℓ [ m ] exists if and only if r ≤ m − ℓ + 1 , wher e as a close family of T yp e II in I ℓ [ m ] exists if and only if r ≤ ℓ + 1 . W e use this opp or tunit y to state the following elemen tary result which comple- men ts Pr op osition 1. This is not stated explic itly in [5, 6 ], but a related r esult is prov ed in [8] where we obtain an alg ebraic counterpart of Prop osition 1 in the setting of exterior algebras and th e Ho dg e star oper ator. Prop ositi on 3 (Dualit y) . Given Λ ⊆ I ℓ [ m ] , let Λ ∗ := { [ m ] \ A : A ∈ Λ } ⊆ I m − ℓ [ m ] . Then Λ is close in I ℓ [ m ] of typ e I if and only if Λ ∗ is close in I m − ℓ [ m ] of typ e II. Pr o of. Giv en S ∈ I ℓ − 1 [ m ] and T ⊆ [ m ] \ S with | T | = r , observe that [ m ] \ ( S ∪ { t } ) = ([ m ] \ ( S ∪ T )) ∪ T \ { t } for ev ery t ∈ T .  As explained in [5 ], Corolla ry 2 essentially ac c ounts for the barrier on r for which the hig her weigh ts d r of Gras smann co des C ( ℓ , m ) ar e hitherto known. (See, e.g., [1 4, 5].) Recently some attempts hav e been made to brea k this ba rrier (cf. [10, 1 1, 8]) but the complete weigh t hierarch y { d r : 1 ≤ r ≤ k } is still not known. W e will commen t more on this in Section 3 . F or the time being, we introduce a combinatorial generalization of close families which may play some role in the determination of higher w eights. Given a subset Λ = { A 1 , · · · , A r } of I ℓ [ m ], w e define K Λ = X i a ≥ 0 4 SUDHIR R. GHORP ADE, AR UNKUMAR R. P A TIL, AND HARIS H K. PILLAI (iii)  a b  b c  =  a c  a − c b − c  . (iv)  a + b c − e  = c X j = e  a + d c − j  b − d j − e  . Pr o of. Both (i) and (ii) are straightforw ard. Pro ofs of (iii) and (iv) a re als o ele- men tary . See, for example, Lemma 3.2 and Co rollar y 3.4 of [4].  The v alue of K r ( ℓ, m ) for the maxim um per missible parameter r is determined below. Prop ositi on 7. K k ( ℓ, m ) = m  ν 2  , wher e ν :=  m − 1 ℓ − 1  . Pr o of. Observe that m ℓ ν =  m ℓ  = k , i.e., ν = ℓk / m . W r ite I ℓ [ m ] = { A 1 , . . . , A k } . Then (1) K k ( ℓ, m ) = X 1 ≤ i 0 for r < m − 1 and m ≥ 4 . The a b ov e result ma y also be compared with one of the cases where equality ho lds in a bound for Σ( G ) obtained by Das [2]. It is well-kno wn that dual graphs G of optimal graphs G [defined in such a wa y that the non-edges of G are the edge s of G ] are optimal (cf. [15, F a ct 1]). This corres po nds precisely to the s p e c ial case ℓ = 2 of the Seco nd Duality Theorem (Prop ositio n 9). F urther , it is ea s y to see that when ℓ = 2, equation (2) co r resp onds precisely to the follo wing elemen tary relation f or simple ( m, r )-g raphs G : Σ( G ) = m ( m − 1) 2 − 4 r ( m − 1) + Σ( G ) . The following b ound, dual to the trivial bound given b y (6), appear s to be new. Corollary 16 . If G is a simple ( m, r ) -gr aph su ch that  m − 1 2  ≤ r ≤  m 2  , then Σ( G ) ≤ m ( m − 1)( m − 2) + ( k − r )( k − r − 1 ) − 4( k − r )( m − 2)+ 2 r, wher e k :=  m 2  . Pr o of. F ollows fro m Corollary 10 and Pr op osition 14.  Finally , we remark that the First Duality Theorem (Prop osition 5) has no ana- logue in the setting of optimal graphs f or the simple reason that it relates optimal graphs to o b jects that ar e not graphs, but h yp er graphs. Indeed, as far a s w e kno w, not m uch seems to b e known ab out threshold h yper graphs and o ptimal h ype r - graphs. Perhaps the notion of a sub clo s e family a nd the results of Section 2 may be o f some help in this dir e c tion. SUBCLOSE F AMILIE S, THRESHOLD GRAPHS, AND LINEAR CODES 11 Ackno wledgments W e are gr ateful to Murali Sriniv a san for helpful discussions and bring ing [15] to our atten tion. References [1] H. Che n, On the mi nimum distance of Sch ub ert codes, IEEE T r ans. Inform. The ory , 46 (2000) 1535–1538. [2] K. C. Das, Maximizing the sum of the squares of the degrees of a graph, Discr ete Math. 285 (2004) 57–66. 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E-mail addr ess : srg@math.i itb.ac.i n Dep ar tment of Ma thema tics, Indian Institute of Technology Bomba y, Pow ai, Mumb ai 400076 , India and Shri Guru Gobind Shing hji Institute of Engineering & Technology, Vishnupuri, Nanded 4 31 606, India E-mail addr ess : arun.iitb@ gmail.co m Dep ar tment of Electrical En gineering, Indian Instit ute of Techn ology Bombay, Pow ai, Mumb ai 400076 , India. E-mail addr ess : hp@ee.iitb .ac.in