Simple proofs for duality of generalized minimum poset weights and weight distributions of (Near-)MDS poset codes
aa r X i v : . [ c s . I T ] A p r JOURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 1
Simple proofs for duality of generalized minimumposet weights and weight distributions of (Near-)MDS poset codes
Dae San Kim,
Member, IEEE,
Dong Chan Kim, and Jong Yoon Hyun,
Abstract —In 1991, Wei introduced generalized minimum Ham-ming weights for linear codes and showed their monotonicity andduality. Recently, several authors extended these results to thecase of generalized minimum poset weights by using differentmethods. Here, we would like to prove the duality by usingmatroid theory. This gives yet another and very simple proofof it. In particular, our argument will make it clear that theduality follows from the well-known relation between the rankfunction and the corank function of a matroid. In addition, wederive the weight distributions of linear MDS and Near-MDSposet codes in the same spirit.
Index Terms —duality, generalized minimum poset weight,weight distribution, MDS poset code, Near-MDS poset code,matroid.
I. I
NTRODUCTION
In 1991, Wei introduced the notion of generalized minimumHamming weights for linear codes [12] and showed theirmonotonicity and duality, motivated by its application tocryptography [9]. Actually, similar properties were consideredearlier for irreducible cyclic codes by Helleseth, Kløve andMykkeltveit in [6].Poset codes were first introduced in [2]. They are justnonempty subsets in F nq , equipped with any poset weightinstead of the usual Hamming weight. By using differentmethods, the duality and monotonicity results were extendedto the case of generalized minimum poset weights for linearposet codes independently by Barg and Purkayastha [1] andde Oliveira Moura and Firer [8]. Later, Choi and Kim [3] alsoshowed the duality for generalized minimum poset weights byexploiting yet another method.Here, we would like to explain very briefly how the dualityresult is proved in each case of [1], [3], and [8]. Barg andPurkayastha in [1], as in the case of Wei’s original proof in[12], do not adopt the matroid theory and exploit instead paritycheck and generator matrices for linear codes. The authorsin [8] adopt the geometric formulation of the generalizedminimum Hamming weights for projective systems in [11]and use multi-set techniques, originated from [5] and [10], inorder to extend the proofs in [11, Theorem 4.1] to the caseof generalized minimum poset weights. So their proof is far D. S. Kim is with the Department of Mathematics, Sogang University,Seoul, Korea. Email: [email protected]. C. Kim is with the Department of Mathematics, Sogang University,Seoul, Korea and the Attached Institute of ETRI, P.O. Box 1, Yuseong,Daejeon, 305-600, Korea. Email: [email protected]. Y. Hyun is with the Department of Mathematics, Ewha WomansUniversity, Seoul 120-750, Korea. Email: [email protected]. different from the original proof of Wei in [12]. Choi andKim in [3] define P ( C ) and RP ( C ) for linear codes C , andshow the duality by using these. In doing so, they obtain moreinformation than just the duality result.The aim of this paper is to present simple proofs for theduality of the generalized minimum poset weights and theweight distributions of linear MDS and Near-MDS poset codesby using only very basic facts of matroid theory [13].In more detail, Theorem 5 is fundamental in proving theduality in Theorem 6 and an analogue of the correspondingTheorem 2 in [12]. One remark here is that while the descrip-tion involving inequality only is given in [12], that involvingboth inequality and equality is stated in our case(cf. (2), (3)).We emphasize here that in showing Theorems 5 and 6 weonly need the facts in Lemma 2, all of which are trivial exceptperhaps (g). It is a special case of (1) applied to the matroid M C of the linear code C , and hence we may say that theduality really follows from the well-known relation betweenthe rank function and the corank function of a matroid. Theweight distributions of linear Near-MDS poset codes wereinvestigated in [1, Theorem 4.1] by using orthogonal array.Here we deduce them in the same spirit as showing the dualitytheorem. Our proof depends on the formula in (4) and needsinformation about the values of the rank (or corank) functionof the associated matroid of linear MDS and Near-MDS posetcodes. For Near-MDS poset codes, we need again the relationbetween the rank function and the corank function of a matroidin order to have that information.II. P RELIMINARIES
The following notations will be used throughout this paper. • F q the finite field with q elements • [ n ] = { , . . . , n } • For J ⊆ [ n ] , J = [ n ] \ J • supp( u ) = { i : u i = 0 } , for u = ( u , . . . , u n ) ∈ F nq • wt H ( u ) = | supp( u ) | the Hamming weight of u • supp( D ) = ∪ u ∈ D supp( u ) , for a subset D ⊆ F nq • P = ([ n ] , ≤ P ) a fixed poset on [ n ] • e P = ([ n ] , ≤ e P ) the dual poset of P on [ n ] (i.e., i ≤ e P j ⇔ j ≤ P i ) • A subset J ⊆ [ n ] is an ideal in P if j ∈ J and i ≤ P j ⇒ i ∈ J • For any J ⊆ [ n ] , h J i P denotes the smallest ideal contain-ing J (i.e., h J i P = { i : i ≤ P j, for some j ∈ J } ) • wt P ( u ) = |h supp( u ) i P | OURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 2 • wt P ( D ) = |h supp( D ) i P | • For J ⊆ [ n ] , u = ( u , . . . , u n ) ∈ F nq , and D ⊆ F nq , u | J = ( u i ) i ∈ J , D | J = { u | J : u ∈ D } • C an [ n, k ] code over F q , with a generator matrix G (an k × n matrix with rank k ) and a parity check matrix H (an ( n − k ) × n matrix with rank n − k ), and with ρ and ρ ⊥ respectively the rank function and the corank function ofthe matriod M C of C . Such a C will be viewed as a linear P -code (i.e., we regard it as a subspace of the P -space ( F nq , wt P ) ) and the dual C ⊥ of C as a linear e P -code • C J = C| J the puncturing of C with respect to J • C J = { u | J : u ∈ C , supp( u ) ⊆ J } the shortening of C with respect to J . Hereafter we will identify C J with thespace l { u ∈ C : supp( u ) ⊆ J } • Φ r ( C ) the set of all r -dimensional subspaces of C , for ≤ r ≤ dim( C ) • Ω( P ) the set of ideals in P • Λ r ( P ) the set of ideals in P of size r • S I = { x ∈ F nq : h supp( x ) i P = I } , for I ∈ Ω( P ) • M ( I ) the set of maximal elements in I , for I ∈ Ω( P ) • I M = I \ M ( I ) , for I ∈ Ω( P ) • Λ( I ) = { J ∈ Ω( P ) : I M ⊆ J ⊆ I } , for I ∈ Ω( P ) • {A r, P ( C ) } nr =0 the P -weight distribution of C with A r, P ( C ) = |{ u ∈ C : wt P ( u ) = r }| A matroid M on S is a finite set S together with afunction(called the rank function of M ) ρ : 2 S → Z ≥ satisfying the following three properties: for A, B ⊆ S ,(R1) ≤ ρ ( A ) ≤ | A | ,(R2) A ⊆ B ⇒ ρ ( A ) ≤ ρ ( B ) ,(R3) ρ ( A ∪ B ) + ρ ( A ∩ B ) ≤ ρ ( A ) + ρ ( B ) .A corank function ρ ∗ is the rank function of the dual matroid M ∗ of M . It is well-known that, for a matroid M withthe rank function ρ and the corank function ρ ∗ , we have thefollowing: for A ⊆ S , ρ ∗ ( S \ A ) = |S| − | A | − ρ ( S ) + ρ ( A ) , or equivalently ρ ∗ ( A ) = | A | − ρ ( S ) + ρ ( S \ A ) . (1)For A ⊆ [ n ] , let G | A and H | A be respectively the submatricesof G and H consisting of the columns indexed by A . Thenwe observe that ρ ( A ) = rank( G | A ) = dim( C| A ) ,ρ ⊥ ( A ) = rank( H | A ) = dim( C ⊥ | A ) . Definition 1.
Let C be an [ n, k ] linear code. For r (1 ≤ r ≤ k ) ,the r -th generalized minimum poset weight( P -weight, if thereference to P is needed) is defined by d P r ( C ) = min { wt P ( D ) : D ∈ Φ r ( C ) } ; for s (1 ≤ s ≤ n − k ) , d e P s ( C ⊥ ) = min { wt e P ( D ) : D ∈ Φ s ( C ⊥ ) } . The following lemma contains all the stuffs that are neededin proving Theorems 2 and 3. Here, all the statements aretrivial except perhaps (g), which is just (1) applied to thematroid M C of C . Lemma 2.
Let J ⊆ [ n ] . Then we have the following. (a) supp( C J ) ⊆ J . (b) For any subset D ⊆ C , supp( D ) ⊆ J ⇔ D ⊆ C J . (c) If J is an ideal in P , then h J i P = J . (d) J is an ideal in P ⇔ J is an ideal in e P . (e) dim( C J ) = dim( C ) − ρ ( J ) . (f) If supp( D ) ⊆ J , for some D ∈ Φ r ( C ) , then ρ ( J ) ≤ dim( C ) − r . (g) | J | − ρ ⊥ ( J ) = dim( C ) − ρ ( J ) = dim( C J ) .Proof: (a), (b), (c), (d) Clear. (e) Let ψ : C → C| J be thelinear map given by u u | J . Then the kernel of this mapis C J . (f) As D ⊆ C J by (b), dim( C J ) ≥ r . The result nowfollows from (e). (g) This follows from (1) and (e). III. P
ROOF OF DUALITY
We do not provide the proof of the following theorem. Onerefers its proof to [1].
Theorem 3.
Let C be an [ n, k ] linear code. Then ≤ d P ( C ) < d P ( C ) < · · · < d P k ( C ) ≤ n, and ≤ d e P ( C ⊥ ) < d e P ( C ⊥ ) < · · · < d e P n − k ( C ⊥ ) ≤ n. Corollary 4.
For ≤ r ≤ k , r ≤ d P r ( C ) ≤ n − k + r. For ≤ s ≤ n − k , s ≤ d e P s ( C ⊥ ) ≤ k + s. Theorem 5.
Let C be an [ n, k ] linear code. For ≤ r ≤ k , d P r ( C ) = min {|h J i P | : | J | − ρ ⊥ ( J ) ≥ r } (2) = min {|h J i P | : | J | − ρ ⊥ ( J ) = r } . (3) For ≤ s ≤ n − k , d e P s ( C ⊥ ) = min {|h J i e P | : | J | − ρ ⊥ ( J ) ≥ s } = min {|h J i e P | : | J | − ρ ⊥ ( J ) = s } . Proof:
Firstly, we show that d P r ( C ) ≤ min {|h J i P | : | J | − ρ ⊥ ( J ) = r } . Let d denote the right hand side of this. Let J be such that | J | − ρ ⊥ ( J ) = r, |h J i P | = d . Then, by Lemma2 (g), dim( C J ) = r . So, d P r ( C ) ≤ wt P ( C J ) ≤ |h J i P | = d ,by Lemma 2 (a). Secondly, we show that min {|h J i P | : | J | − ρ ⊥ ( J ) ≥ r } ≤ d P r ( C ) . Let e denote the left hand side of this.To show this, let wt P ( D ) = d P r ( C ) , for some D ∈ Φ r ( C ) .Set J = h supp( D ) i P . Then D ⊆ C J , by Lemma 2 (b) and dim( C J ) = | J |− ρ ⊥ ( J ) ≥ r (cf. Lemma 2 (g)). So, by Lemma2 (c), e ≤ |h J i P | = | J | = d P r ( C ) . Lastly, it is enough to seethat d ≤ e . Let e = |h J i P | , with | J | − ρ ⊥ ( J ) ≥ r . Thenwe claim that | J | − ρ ⊥ ( J ) = r . Assume on the contrary that | J | − ρ ⊥ ( J ) = r ′ > r . Then, by the first and second steps, OURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 3 d P r ′ ( C ) ≤ min {|h I i P | : | I | − ρ ⊥ ( I ) = r ′ } ≤ |h J i P | = e ≤ d P r ( C ) , a contradiction to Theorem 3. Theorem 6.
Let C be an [ n, k ] linear code and A = { d P r ( C ) :1 ≤ r ≤ k } , B = { n + 1 − d e P s ( C ⊥ ) : 1 ≤ s ≤ n − k } . Then A and B are disjoint and [ n ] = A ∪ B .Proof: It is enough to see that A and B are disjoint.Let s be any integer such that ≤ s ≤ n − k . Then weneed to see that n + 1 − d e P s ( C ⊥ ) A . Firstly, let t = k + s − d e P s ( C ⊥ ) By Corollary 4, s ≤ d e P s ( C ⊥ ) ≤ k + s , so that ≤ t ≤ k . Then we claim that d P t ( C ) ≤ n − d e P s ( C ⊥ ) , sothat n + 1 − d e P s ( C ⊥ ) = d P r ( C ) , for r ≤ t . Let wt e P ( D ) = d e P s ( C ⊥ ) , for some D ∈ Φ s ( C ⊥ ) . Set I = h supp( D ) i e P . Then D ⊆ ( C ⊥ ) I , by Lemma 2 (b), and hence, by Lemma 2 (g), | I | − ρ ( I ) = dim(( C ⊥ ) I ) ≥ dim( D ) = s . So, by Lemma 2(g), ρ ⊥ ( I ) = | I |− k + ρ ( I ) = ( n − k ) − ( | I |− ρ ( I )) ≤ n − k − s ,and, with J = I, | J | − ρ ⊥ ( J ) ≥ ( n − | I | ) − ( n − k − s ) = k + s − d e P s ( C ⊥ ) = t . So, d P t ( C ) ≤ |h J i P | = | J | = n − d e P s ( C ⊥ ) , byTheorem 5 and as J is an ideal in P by Lemma 2 (d). Secondly,we must show that n + 1 − d e P s ( C ⊥ ) = d P t +1 ( C ) , for all l with ≤ l ≤ k − t . Suppose that n +1 − d e P s ( C ⊥ ) = d P t + l ( C ) , for some l . Then, for some D ∈ Φ t + l ( C ) , let wt P ( D ) = d P t + l ( C ) = n +1 − d e P s ( C ⊥ ) . If J = I , with I = h supp( D ) i P , then, by Lemma2 (f), ρ ( J ) ≤ k − t − l . So | J |− ρ ( J ) ≥ ( n −| I | ) − ( k − t − l ) =( d e P s ( C ⊥ ) − − ( d e P s ( C ⊥ ) − l − s ) = s + l − . By Lemma 2(d), J is an ideal in e P , and, by Theorem 5 and Lemma 2(c), d e P s + l − ( C ⊥ ) ≤ |h J i e P | = | J | = d e P s ( C ⊥ ) − , which is acontradiction to Theorem 3.IV. W EIGHT DISTRIBUTIONS OF LINEAR
MDS
AND N EAR -MDS
POSET CODES
The equation (4) in the following follows from [7, (3.1)],while the equation (5) is clear.
Proposition 7.
Let I be an ideal in P . (a) |C ∩ S I | = X J ∈ Λ( I ) ( − | I |−| J | q k − ρ ( J ) . (4)(b) A r, P ( C ) = X I ∈ Λ r ( P ) |C ∩ S I | . (5)In what follows, we will denote d P ( C ) simply by d . Let C be a MDS P -code with parameters [ n, k, d = n − k + 1] . Thenone easily shows from [1, Lemma 2.2 (4)] that, for J ∈ Ω( P ) , k − ρ ( J ) = (cid:26) , | J | ≤ d − , | J | − d + 1 , | J | ≥ d . So, for any I ∈ Λ r ( P ) , from (4) we have |C ∩ S I | = X J ∈ Λ( I ) ( − r −| J | q k − ρ ( J ) = X J ∈ Λ( I ) | J |≤ d − ( − r −| J | + X J ∈ Λ( I ) | J |≥ d ( − r −| J | q | J |− d +1 = X J ∈ Λ( I ) ( − r −| J | + X J ∈ Λ( I ) | J |≥ d ( − r −| J | ( q | J |− d +1 − . (6)Now, the first sum in (6) is X J ∈ Λ( I ) ( − r −| J | = r X l = | I M | X J ∈ Λ( I ) | J | = l ( − r − l = r X l = | I M | ( − r − l X J ∈ Λ( I ) | J | = l r X l = | I M | ( − r − l (cid:18) | M ( I ) | l − | I M | (cid:19) = r −| I M | X s =0 ( − r −| I M |− s (cid:18) | M ( I ) | s (cid:19) = ( − | M ( I ) | | M ( I ) | X s =0 ( − s (cid:18) | M ( I ) | s (cid:19) = 0 . (7)The second sum in (6) is X J ∈ Λ( I ) | J |≥ d ( − r −| J | ( q | J |− d +1 − r − d X l =0 X J ∈ Λ( I ) | J | = d + l ( − r − d − l ( q l +1 − r − d X l =0 ( − r − d − l ( q l +1 − X J ∈ Λ( I ) | J | = d + l r − d X l =0 ( − r − d − l ( q l +1 − (cid:18) | M ( I ) | r − d − l (cid:19) = r − d X s =0 ( − s ( q r − d +1 − s − (cid:18) | M ( I ) | s (cid:19) . (8)Thus we obtain the following theorem from (5)-(8). Theorem 8.
Let C be a MDS P -code with parameters [ n, k, d = n − k + 1] . Then, for r , with d ≤ r ≤ n , A r, P ( C ) = X I ∈ Λ r ( P ) r − d X s =0 ( − s (cid:18) | M ( I ) | s (cid:19) ( q r − d +1 − s − . Recall that an [ n, k ] P -code is called a Near-MDS P -codeif d = d P ( C ) = n − k, d P ( C ) = n − k + 2 . OURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 4
Lemma 9 ([1, Lemma 2.4 (1), (2)]) . The following hold. (a) C is an [ n, k ] Near-MDS P -code if and only if (1) Any n − k − columns of the parity check matrix H are linearly independent. (2) There exist n − k linearly dependent columns of H . (3) Any n − k + 1 columns of H have the full rank n − k . (b) If C is a linear Near-MDS P -code, then C ⊥ is a linearNear-MDS e P -code. Now, we assume that C is a Near-MDS P -code withparameters [ n, k, d = n − k ] . Then, from Lemma 9 (a) above,we get ρ ⊥ ( J ) = (cid:26) | J | , | J | < n − k , n − k, | J | > n − k . (9)By invoking Lemma 2 (g) again, from (9) we have, for J ∈ Ω( P ) , k − ρ ( J ) = (cid:26) , | J | ≤ d − , | J | − d, | J | ≥ d + 1 . (10)We note here that (10) also follows from (9) and Lemma 9(b). However, Lemma 9 (b) is deduced in [1] from the dualityresult in Theorem 6, which in turn follows from Lemma 2 (g),as we stressed in Section I.Then, by proceeding analogously to the MDS case, we getthe following result. Theorem 10 ( [1]) . Let C be a Near-MDS P -code withparameters [ n, k, d = n − k ] . Then, for r , with d ≤ r ≤ n , A r, P ( C ) = X I ∈ Λ r ( P ) r − d − X s =0 ( − s (cid:18) | M ( I ) | s (cid:19) ( q r − d − s − − r − d X I ∈ Λ r ( P ) X J ∈ Λ( I ) | J | = d A J ( C ) , (11) where A J ( C ) = |C ∩ S J | . In the case of Hamming weight(i.e., wt P with P the an-tichain on [ n ] ), denoting A r, P ( C ) by A r ( C ) as usual, werecover the following corollary in [4]. Corollary 11 ( [4]) . Let C be a Near-MDS code with param-eters [ n, k, d = n − k ] . Then, for r , with d ≤ r ≤ n , A r ( C ) = (cid:18) nr (cid:19) r − d − X s =0 ( − s (cid:18) rs (cid:19) ( q r − d − s − − r − d (cid:18) n − dr − d (cid:19) A d ( C ) , Proof: Now, let P denote the antichain. Then the seconddouble sum in (11) is X | I | = r X u ∈C w H ( u )= d supp( u ) ⊆ I (cid:18) n − dr − d (cid:19) A d ( C ) . by counting I , with | I | = r , for each fixed u ∈ C , with w H ( u ) = d . R EFERENCES[1] A. Barg and P. Purkayastha,
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Dae San Kim(M’05) received the B.S. and M. S. degrees in mathematics fromSeoul National University, Seoul, Korea, in 1978 and 1980, respectively, andthe Ph.D. degree in mathematics from University of Minnesota, Minneapolis,MN, in 1989. He is a professor in the Department of Mathematics atSogang University, Seoul, Korea. He has been there since 1997, following aposition at Seoul Women’s University. His research interests include numbertheory(exponential sums, modular forms, zeta functions) and coding theory.
Dong Chan Kim received the B. S. and M. S. degrees in mathematicsfrom Sogang University, Seoul, Korea, in 2001 and 2003, respectively, and iscurrently pursuing a Ph. D. degree in mathematics at Sogang University. Hehas been working as a researcher at the Attached Institute of ETRI.