SIMEON BALL AND AART BLOKHUIS
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1 A BOUND FOR THE MAXIMUM WEIGHT OF A LINEAR CODE SIMEON BALL AND AART BLOKHUIS Absrac. I s shown ha h paramrs of a lnar cod ovr F q of lngh n, dmnson k, mnmum wgh d and maxmum wgh m sasfy a cran congrunc rlaon. In h cas ha q = p s a prm, hs lads o h bound m (n dp (p 1, whr {0, 1,..., k 2} s maxmal wh h propry ha ( n d 0 (mod p k 1. Thus, f C conans a codword a lngh n hn n d/(p 1 + d Inroducon A lnar cod C s a k-dmnsonal subspac of F n q. W say ha C has lngh n and dmnson k. Th wgh of a vcor s h numbr of non-zro coordnas and w dno by d h mnmum non-zro wgh of C. Th Hammng dsanc bwn wo vcors u and v s h numbr of coordnas n whch hy dffr. I s a smpl mar o obsrv ha h mnmum dsanc bwn any wo vcors of C s d, h mnmum wgh of C. For mor on lnar cods, s [6] or [7]. L q = p h, whr p s a prm. In hs arcl w shall prov ha f hr s a lnar cod of lngh n, dmnson k, mnmum dsanc d and maxmum wgh m n d + 1 hn for all ɛ 1, h coffcn of X (n dq m+ɛ n s dvsbl by q k 1. (1 + X m (1 + X p (n d Da: 1 Jun Ths rsarch was nad whl h auhors wr vsng h Insu for Mahmacal Scncs, Naonal Unvrsy of Sngapor n Th frs auhor acknowldgs h suppor of h projc MTM C03-01 of h Spansh Mnsry of Scnc and Educaon and h projc 2009-SGR of h Caalan Rsarch Councl. 1
2 2 BALL AND BLOKHUIS For q = p hs gvs h bound m (n dp (p 1, whr {0, 1,..., k 2} s maxmal wh h propry ha ( n d 0 (mod p k 1. Hnc, f C conans a codword of wgh n hn n d/(p 1 + d +. Ths s probably bs compard wh h Grsmr bound ([4] whch sas ha for a lnar cod of lngh n, dmnson k and mnmum dsanc d ovr F q, k 1 n d/q. =0 In Scon 6 w ransla h bounds oband o bounds for (n, -arcs and -fold blockng ss of hyprplans n AG(k 1, q, h Dsargusan affn spac. W shall us som vry basc proprs of complx characrs, whch w rvw n h nx scon. and dno by Ĝ = {χ u u G} h mul- L G b h addv group of F k 1 q plcav group of characrs, so 2. Group characrs χ u (x = 2πTr(x u/p, whr q = p h for som prm p, Tr s h rac funcon from F q o F p and x u s h sandard nnr produc. Lmma 2.1. L g(x = χ Ĝ c χχ(x, whr c χ Z. If g(x = 0 for all x G\{0}, hn q k 1 dvds g(0. Proof. W hav g(0 = g(x = c χ χ(x = x G x G χ Ĝ χ Ĝ c χ χ(x = c χ0 G, x G snc x G χ(x = 0 unlss χ = χ 0 n whch cas s G.
3 MAXIMUM WEIGHT OF A LINEAR CODE 3 3. Lnar cods conanng a codword of wgh qual o s lngh L C b a lnar cod of lngh n, dmnson k and mnmum dsanc d whch conans a codword of wgh n. L A b a k n gnraor marx for C, so ha C = {xa x F k q}, whos k-h row s a codword of wgh n. L S b h mul-s of n vcors of F k q whch ar h columns of A. For any x F k q, h vcor xa has a las d non-zro coordnas and so has a mos n d zro coordnas. Hnc, hr ar a mos n d vcors n S wh h propry ha x 1 s x k s k = 0. Thrfor, hr ar a mos n d vcors n S on h hyprplan of F k q dfnd by h quaon x 1 X x k X k = 0. Mulplyng u S by a non-zro scalar dos no affc hs propry, so w can assum ha h k-h coordna of ach of h vcors n S s 1. Thus, w can consdr S as a subs of AG(k 1, q. As w hav sn, vry hyprplan of AG(k 1, q conans a mos n d pons of S. Now, fx an x F k 1 q, x 0 and consdr h q hyprplans of AG(k 1, q dfnd by h quaon x 1 X x k 1 X k 1 = α, whr α F q. Each of hs hyprplans conans a mos n d pons of S, whch has sz n, and so n (n dq. Th followng horm suggss ha n gnral hr s a much br bound. Thorm 3.1. L C b a lnar cod of lngh n, dmnson k and mnmum dsanc d ovr F q, whr q = p h and p s prm. If C conans a codword of wgh n hn, for all ɛ 1 and γ n d, h coffcn of X γq n+ɛ n s dvsbl by q k 1. (1 + X n (1 + X p γ Proof. L f(x, x = u S(X + χ u (x,
4 4 BALL AND BLOKHUIS so ha s a polynomal whos coffcns ar complx valud funcons, and for vry x G hs dfns a polynomal f(x, x C[X]. L g(x, x = g j (xx j j=0 b dfnd by f(x, xg(x, x = 1 and no ha g j (x = c χ χ(x, χ Ĝ for som c χ Z. Furhrmor, for som γ n d, dfn and so h(x, x = (X p + 1 γ g(x, x f(x, xh(x, x = (X p + 1 γ. For x 0 Fq k 1, x 0 0, and α F q, hr ar a mos n d pons u S such ha x 0 u = α. Thus, h mul-s {χ u (x 0 u S} conans ach p-h roo of uny rpad a mos (n d ms. Ths mpls ha f(x, x 0 dvds (X p + 1 γ and so h(x, x 0 s a polynomal, and s a polynomal of dgr γq n. Thrfor, for all ɛ 1, h coffcn of X γq n+ɛ n h(x, x 0 s zro. Hnc, h funcon s zro, for all x = x 0 0. By Lmma 2.1, (γq n+ɛ/p r=0 (γq n+ɛ/p r=0 ( γ r ( γ r g γq n+ɛ rp (x g γn n+ɛ rp (0 = 0 (mod q k 1, and so h coffcn of X γq n+ɛ n h(x, 0 s dvsbl by q k 1. I only rmans o no ha snc f(x, 0 = (X + 1 n, w hav h(x, 0 = (1 + X n (1 + X p γ.
5 MAXIMUM WEIGHT OF A LINEAR CODE 5 4. A condon on h paramrs of a lnar cod Thorm 3.1 has h followng corollary. Corollary 4.1. L C b a lnar cod of lngh n, dmnson k, mnmum dsanc d and maxmum wgh m ovr F q, whr q = p h and p s prm. If m n d + 1 hn, for all ɛ 1 and γ n d, h coffcn of X γq m+ɛ n s dvsbl by q k 1. (1 + X m (1 + X p γ Proof. Th cod C shorns o a cod of lngh m, dmnson k and mnmum dsanc d, whr m d n d conanng a codword of lngh m. Apply Thorm 3.1 o h shornd cod. W nroduc a sum π l,m whch wll allow us o xplo hs congrunc. L b h s of p-h roos of uny and dfn π l,m = δ (1 + δx m (1 + X p l. Th coffcn of X rp n π n d,m s p ms h coffcn of X rp n (1 + X m (1 + X p (n d, for all r N. Thrfor, f w can calcula h xac numbr of ms p dvds h coffcns of π n d,m hn w can us Corollary 4.1 o oban a bound for m. W wll us h followng lmma. Lmma 4.2. For l 1, π l,m = q 1 =1 = /p ( ( p Proof. For all δ, usng h bnomal horm, ( (1 + X p = X p = =0 ( 1 + π l 1,m + π l 1,m q. =0 ( (1 (1 + δx p ( 1
6 6 BALL AND BLOKHUIS = =0 p =0 ( ( p ( 1 + (1 + δx = No ha 0 = (1 + ( 1 = =0 Snc π l,m = δ (1 + X p = (1 + δx q + ( q 1 q =0 = /p ( ( p ( 1, so w hav ha =1 = /p ( ( p ( 1 + (1 + δx. ( 1 + (1 + δx (1 + δx m (1 + X p l = δ (1 + δx m (1 + X p (l 1 (1 + X p, h lmma follows. W can rpadly apply Lmma 4.2, rducng h frs subndx by on and rducng h scond subndx succssvly by 1, 2,..., l. By sng s o b h numbr of hs j s ha ar no qual o q w can wr hs rducon as n h followng lmma. Lmma 4.3. n d ( n d π n d,m = s s=0 1,..., s {1,...,q 1} c 1... c s π 0,m ( s (n d sq, whr c = = /p ( ( p ( 1 +. In gnral sms dffcul o calcula h xac numbr of ms p dvds c and so b abl o apply Corollary 4.1. Howvr, n h cas q = p hs s asly don, snc for 0 < < q = p, ( p c = ( Ths s wha w shall us n h followng scon.
7 MAXIMUM WEIGHT OF A LINEAR CODE 7 5. Lnar cods ovr a prm fld Thorm 5.1. L C b a lnar cod of lngh n, dmnson k, mnmum dsanc d and maxmum wgh m ovr F p, whr p s prm. If m n d + 1 hn m (n dp (p 1, whr {0, 1,..., k 2} s maxmal wh h propry ha ( n d 0 (mod p k 1. Proof. By shornng h cod C f ncssary, whch may possbly dcras n d, w can suppos ha m = (n dp (p By Corollary 4.1, for all ɛ 1 h coffcn of X (p 1+ɛ 1 n s zro modulo p k 1. (1 + X m (1 + X p n d, As mnond bfor, h coffcn of X rp n π n d,m s p ms h coffcn of X rp n (1 + X m (1 + X p n d, for all r N. Choos ɛ so ha p dvds (p 1 + ɛ 1 and no ha h coffcn of X (p 1+ɛ 1 n π n d,m s zro modulo p k. Consdr h rms n h sum n Lmma 4.3. If m ( s (n d sp 0 hn π 0,m (1 + + s (n d sp s a polynomal and morovr s a polynomal of dgr s (p 1 sp 1 + j (p 1 s 1, j=1 and so has no rm of dgr (p 1 + ɛ 1. If s 1 hn m ( s (n d sp < 0 and so π 0,m (1 + + s (n d sp has no rm of dgr (p 1 + ɛ 1. If s + 1 hn snc p dvds c j for all j = 1,..., s, ( n d s = 0 mod p k 1 s by hypohss, and all coffcns of π 0,m (1 + + s (n d sp ar dvsbl by p, all rms n h sum n Lmma 4.3 ar zro modulo p k.
8 8 BALL AND BLOKHUIS If s = and m ( s (n d sp 1 hn ( s + s 0 and so j = 1 for all j = 1,..., s. Thus, h coffcn of X (p 1 1+ɛ n π n d,m s h coffcn of X (p 1 1+ɛ n ( n d c 1π 0,1, and snc c 1 = p and h coffcn of X (p 1 1+ɛ n π 0,1 s p( 1 (p 1 1+ɛ, s no zro modulo p k, a conradcon. 6. (n, r-arcs and -fold blockng ss of AG(s, q An (n, -arc n AG(s, q s a s S of pons wh h propry ha any hyprplan conans a mos pons of S. Rvrsng h consrucon of Scon 3, h cod gnrad by h (s + 1 n marx whos columns ar h vcors n S and whr h (s + 1-h row s h all on vcor s lnar cod of lngh n, dmnson s + 1 and mnmum dsanc a las n. Thrfor, Corollary 4.1 has h followng corollary. Corollary 6.1. If hr s an (n, -arc n AG(s, q, whr q = p h and p s prm, hn for all ɛ 1 h coffcn of X q n+ɛ n s dvsbl by q k 1. (1 + X n (1 + X p And Thorm 5.1 has h followng corollary. Corollary 6.2. If hr s an (n, -arc n AG(s, p, whr p s prm, hn n ( p +, whr {0, 1,..., s 1} s maxmal wh h propry ha ( 0 (mod p s. A -fold blockng s of hyprplans n AG(s, q s a s B of pons of AG(s, q wh h propry ha vry hyprplan conans a las pons of B. Th complmn of B s a (q s B, q s 1 -arc of AG(s, q. Hnc, h abov corollars mply h followng for -fold blockng ss.
9 MAXIMUM WEIGHT OF A LINEAR CODE 9 Corollary 6.3. If B s a -fold blockng s of AG(s, q, whr q = p h and p s prm, hn for all ɛ N = {1, 2,...} h coffcn of X B q+ɛ n s dvsbl by q k 1. (1 + X B qs (1 + X p (qs 1 And n h cas ha q s prm w hav h followng corollary. Corollary 6.4. If B s a -fold blockng s of AG(s, p, whr p s prm, hn B p + (p 1, whr {0, 1,..., s 1} s maxmal wh h propry ha ( 0 (mod p s. Ths bound should b compard wh h followng bounds. Th bound of Brun [3] B ( + s 1(q 1 + 1, s a gnral lowr bound, whch mprovs on h rval B q for (s 1(q 1. Ths bound had bn oband prvously for = 1 by Jamson [5] and Brouwr and Schrjvr [2]. Th bound from [1], whch was provn hr for < q, sas ha provdd ha B q + (s 1(q 1 ( s 0 1 (mod p. For many paramrs Corollary 6.3 wll allow on o calcula a br lowr bound han hs prvously known bounds, as Corollary 6.4 ndcas n h cas q s prm. Indd, for q prm h bound B q + (s 1(q 1 xnds o all provdd ha ( ( s = ( 1 s+ s 1 1 whch mprovs on Brun s bound by 1. 0 (mod p,
10 10 BALL AND BLOKHUIS Rfrncs [1] S. Ball, On nrscon ss n Dsargusan affn spacs, Europan J. Combn., 21 ( [2] A. E. Brouwr and A. Schrjvr, Th blockng numbr of an affn spac, J. Combn. Thory Sr. A, 24 ( [3] A. A. Brun, Polynomal mulplcs ovr fn flds and nrscon ss, J. Combn. Thory Sr. A, 60 ( [4] J. H. Grsmr, A bound for rror-corrcng cods, IBM J. Rs. Dvlop., 4 ( [5] R. Jamson, Covrng fn flds wh coss of subspacs, J. Combn. Thory Sr. A, 22 ( [6] F. J. MacWllams and N. J. A. Sloan, Th Thory of Error-Corrcng Cods, Norh-Holland, [7] J. H. van Ln, An Inroducon o Codng Thory, Thrd don, Sprngr-Vrlag, Smon Ball Dparamn d Mamàca Aplcada IV, Unvrsa Polècnca d Caalunya, Jord Grona 1-3, Mòdul C3, Campus Nord, Barclona, Span smon@ma4.upc.du Aar Blokhus Dparmn of Mahmacs and Compung Scnc, Endhovn Unvrsy of Tchnology, P.O. Box MB Endhovn, Th Nhrlands aarb@wn.u.nl
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