New Upper Bounds on A(n,d)

Transcription

1 New Upper Bouds o A(d Beam Mouts Departmet of Mathematcs Techo-Israel Isttute of Techology Hafa 3000 Israel Emal: moutsb@techuxtechoacl Tu Etzo Departmet of Computer Scece Techo-Israel Isttute of Techology Hafa 3000 Israel Emal: etzo@cstechoacl Smo Ltsy Departmet of Electrcal Egeerg Tel A Uersty Ramat-A Israel Emal: ltsy@egtauacl arx:cs/ [csit] 4 Aug 005 Abstract Upper bouds o the maxmum umber of codewords a bary code of a ge legth mmum Hammg dstace are cosdered New bouds are dered by a combato of lear programmg coutg argumets Some of these bouds mproe o the best kow aalytc bouds Seeral ew record bouds are obtaed for codes wth small legths I INTRODUCTION Let A(d deote the maxmum umber of codewords a bary code of legth mmum Hammg dstace d A(d s a basc quatty codg theory Lower bouds o A(d are obtaed by costructos For surey o the kow lower bouds the reader s referred to [9] I ths work we cosder upper bouds o A(d The most basc upper boud o A(d d = e1 s the sphere packg boud also kow as the Hammg boud: A(e1 (1 Johso [8] has mproed the sphere packg boud I hs theorem Johso used the quatty A(dw whch s the maxmum umber of codewords a bary code of legth costat weght w mmum dstace d: A(e1 I [11] a ew boud was obtaed: A(e1 e1 ( e1 e1a(ee1 A(ee1 ( 1 e ( e ea(1ee A(1ee (3 Ths boud s at least as good as the Johso boud for all alues of d for each d there are ftely may alues of for whch the ew boud s better tha the Johso boud Whe someoe s ge specfc relately small alues of d usually the best method to fd upper boud o A( d s the lear programmg (LP boud A summary about ths method some ew upper bouds appeared [11] Howeer the computato of ths boud s ot tractable for large alues of I ths work we wll preset ew upper bouds o A(e1 e 1 Let F = {01} let F deote the set of all bary words of legth For xy F d(xy deote the Hammg dstace betwee x y Ge xy F such that d(xy = k we deote by p k j the umber of words z F such that d(xz = d(zy = j Ths umber s depedet of choce of x y equal to ( k k jk j k f j k s ee p k j = 0 f j k s odd If x = y the p 0 j = δ j = s the umber of words at dstace from x F δ j = 1 f = j zero otherwse We also deote = The p k j s are the tersecto umbers of the Hammg scheme s the alecy of the relato R For the coecto betwee assocato schemes codg theory the reader s referred to [6] [10 Chapter 1] A (Me1 code C s a oempty subset of F of cardalty M mmum Hammg dstace e1 For a wordx F d(xc s the Hammg dstace betweex C e d(xc = m c C d(xc A word h F s called a hole f d(hc > e H = {h F : d(hc > e} s the set of all holes Clearly we hae H = C V(e (4 V(e = j t s the olume of sphere of radus e The dstace dstrbuto of C s defed as the sequece A = {(c 1 c C C : d(c 1 c = } / C for 0 A (c deote the umber of codewords at dstace from c C We also defe the (o-ormalzed holes dstace dstrbuto{d } byd = {(h 1 h H H : d(h 1 h = } D (h deote the umber of holes at dstace from h H Fally we defe NC(hC to be the umber of codewords of C at dstace from a hole h II HOLES DISTANCE DISTRIBUTION I the frst theorem we state that for a ge (Me 1 code C the dstace dstrbuto of the holes s uquely determed by the dstace dstrbuto {A } of the code C Theorem 1: If C s a (Me1 code wth dstace dstrbuto {A } the D = C (R(C V(e

2 for each 0 R(C = δ k p k j k=0 j=1 p k lm p l j l=1 m=1 j=1 A k Corollary 1: Let C be a (Me1 code wth dstace dstrbuto{a } If {q } s a sequece of real umbers the q D = q C q (R(C V(e By usg Corollary 1 for ay ge sequece {q } we obta a lear combato of the D s By fdg a lower boud o ths combato we ca obta a upper boud o the sze of C Example 1: Let q 0 = 1 q = 0 > 0 Clearly 0 = 1 by (5 we hae R(C0 = V(e After substtutg the tral boud D 0 0 to (6 we obta the sphere packg boud (1 The sequece {q } of Corollary 1 wll be called the holes dstace dces (HDI sequece For coeece the rest of the paper we wll wrte {q } stead of {q } I the ext two sectos we wll fd some good HDI sequeces {q } deelop methods to fd lower bouds o q D III HDI SEQUENCES WITH SMALL INDICES I ths secto we cosder HDI sequeces ozero q s correspod to small dces The followg lemma ges a alterate expresso for D Lemma 1: For each 0 D = e k=e1 NC(hCk p k j Ge a sequece {q } by usg Lemma 1 (4 we estmate q D the followg way q D = = q q e k=e1 e q k=e1 NC(hCk NC(hCk p k j p k j (5 (6 ( C V(e q ξ(c{q } (7 ξ(c{q } = max e q k=e1 By combg (6 (7 we obta NC(hCk p k j (8 Theorem : If C s a (Me1 code wth dstace dstrbuto {A } the V(e q(v(e R(C ξ(c{q } proded ξ(c{q } s ot zero ξ(c{q } s ge by (8 R(C s ge by (5 Example : Let q 1 = 1 q = 0 for 1 From (5 (8 we hae R(C1 = V(e 1 p e1 1e e1 p e1 1e pe1 e1e A e1 ξ(c{q } = p e1 1e max {NC(hCe1} Thus usg Theorem we obta Theorem 3: If C s a (Me1 code wth dstace dstrbuto {A } the By substtutg V(e e1 pe1 e1e Ae1 max {NC(hCe1} A e1 A(ee1 max{nc(hce1} A(ee1 (9 we obta the Johso upper boud ( Example 3: Let q 1 = pe e pe1 e 1 pe1 e p e1 1e q = 1 q = 0 for / {1} From (5 (8 we hae (9 q 1 R(C1q R(C = V(e(q 1 1 q p e e ( e1 e (p e1 e1e pe1 ee 1 pe1 ee A e1 p e ee A e ξ(c{q } = p e e max {NC(hCe1NC(hCe} Thus usg Theorem we obta

3 Theorem 4: If C s a (Me 1 code wth dstace dstrbuto {A } the V(e e1 e γ max {NC(hCe1NC(hCe} (10 γ = (p e1 e1e pe1 ee 1 pe1 ee A e1 p e ee A e By substtutg A e1 A e A(1ee max {NC(hCe1NC(hCe} A(1ee (10 we obta the boud of (3 Next we wat to mproe the tral boud o A ge by A A(e We wll fd upper bouds o dstace dstrbuto coeffcetsa s usg lear programmg For a (Me1 code C wth dstace dstrbuto {A } let us deote by LP[e1] the followg system of Delsarte s lear costrats: A P k ( 0 for 0 k 0 A A(e for = e1e A 0 = 1A = 0 for 1 < e1 P k ( = k ( 1j( j k j deote Krawtchouk polyomal of degree k We also deote ñ = 1 let {Ã}ñ be the dstace dstrbuto of the (1Me exteded code C e whch s obtaed from the (Me1 code C wth dstace dstrbuto {A } by addg a ee party bt to each codeword of C It s easy to erfy that for each e1 ñ/ à = A 1 A (11 For the ee weght code C e of legth ñ dstace d = e we deote by LP e [ñe] the followg system of Delsarte s lear costrats: ñ ÃP k ( 0 for 0 k ñ/ 0 à A(ñd for = ee4 ñ/ à 0 = 1à = 0 for 1 < e I some cases we wll add more costrats to obta some specfc bouds as [5] [7] [11] [1] By Theorem 4 we hae that for a (Me1 code C wth dstace dstrbuto {A } the followg holds: Usg (11 we obta 1 e ( e e(a e1a e A(1ee Theorem 5: A(e1 1 e ( e emax{ãe} A(1ee max{ãe} s take subject to LP e [ñe] For the ext result we eed the followg theorem whch s a geeralzato of a theorem ge by Best [3] Theorem 6: Let C be a code of legth mmum Hammg dstace d dstace dstrbuto {A } Let {p } be a sequece of real umbers The there exsts a code C of legth 1 dstace d wth dstace dstrbuto {A } 1 satsfyg 1 ( p A p A (1 It was proed [13] by usg LP that for a ee weght code C of legth ñ 1(mod 4 dstace d = 4 dstace dstrbuto {Ã}ñ à 4 (ñ 1(ñ (ñ 3 (13 4 We substtute p = δ 4 (1 (13 for the upper boud o A 4 to obta Lemma : If C s a ee weght code of legth ñ (mod 4 dstace d = 4 dstace dstrbuto {Ã}ñ the à 4 ñ(ñ (ñ 3 4 We take e = 1 9(mod 1 Sce A(143 = ( 3/6 for 9(mod 1 [10 p 59] t follows by Lemma Theorem 5 that Theorem 7: For 9(mod 1 A( The preous best kow boud A(3 /( 3 for 1(mod 4 was obtaed [4] by LP I partcular we hae A( whch mproes o the preous best kow boud A( [11] IV HDI SEQUENCES WITH LARGE INDICES We demostrate aother approach to estmatg q D ozero elemets of {q } correspod to large dces For each t 0 t e we deote E t = {h H NC(hC t = 1} Note that for ay hole h H we hae NC(hC t {01} 0 t e

4 Lemma 3: For each t 0 t e et E t = C t A p tj (14 Let q 1 = q = 1 q = 0 for / { 1} If h E t for t {01e 1} the If h E e the D 1 (hd (h = 0 (15 D 1 (hd (h e (e1a( eee1 = e (e1 e (16 e1 If for a ge hole h there exsts o codeword at dstace k { e (e 1 1} the D 1 (hd (h 1 (e1a(ee1 = 1 (e1 (17 e1 By combg (14-(17 wth Corollary 1 we obta Theorem 8: A(e1 e(( e (e1 e U( = max{r(c 1R(C ( e 1 et 1 (e1 e1 t=0 ( (e1 1 e e1 e e1 e1 U( (e1 e1 A A p tj p ej } subject to LP[e1] R(C 1 R(C are ge by (5 By Theorem 8 we obta A( A( whch mproe the preous best kow bouds A( [11] A( [14] Let be ee teger let e = 1 By Theorem 8 (11 we obta Theorem 9: If s a ee teger the subject to LP e [ñ4] A(3 3 U( U( = max{6ãñ 3 3Ãñ 1} Usg LP we ca proe the followg lemma Lemma 4: If C s a ee weght code of legth ñ 11(mod 1 dstace d = 4 dstace dstrbuto {Ã}ñ the 6Ãñ 3 3(ñ 1Ãñ 1 (ñ 1(ñ (ñ4 ñ Therefore by Theorem 9 Lemma 4 we hae Theorem 10: For 10(mod 1 A(3 8 3 The preous best kow aalytc boud A(3 (14 8 was obtaed by (3 By smlar argumets f {q } s a sequece wth q = 0 except for q = q 1 = q = 1 we obta the followg boud Theorem 11: If C s a (Me1 code the φ = ( φ U( ( e A(1ee 1 e (1 e ( e A(1 eee ( e U( = max{ A(1ee = ( 1 R(C (1 ( e e et A(1ee A ( e1 (1e( e A(1 eee ( e t=0 p tj (A(1ee t=e 1 et A p tj } subject to LP[ e1] R(C R(C 1 R(C are ge by (5 Applyg Theorem 11 we obta A( whch s better tha the best preously kow boud (see Secto III

5 V GENERALIZATION FOR ARBITRARY METRIC ASSOCIATION SCHEMES We ca geeralze our approach to arbtrary metrc assocato scheme (X R wth dstace fucto d whch cossts of a fte set X together wth a set R of1 relatos defed o X wth certa propertes For the complete defto bref troducto to the assocato schemes the reader s referred to [10 Chapter 1] We exted the deftos from the frst secto as follows X = s the umber of pots of a fte set X s the alecy of the relato R p k j s are the tersecto umbers of the scheme A code C s a oempty subset of X wth mmum dstace e 1 The deftos related to holes dstace dstrbuto are easly geeralzed The results of (4 Lemma 1 Theorems 1 through 4 Corollary 1 are ald for arbtrary metrc assocato schemes As a example we cosder the Johso scheme I ths scheme X s the set of all bary ectors of legth weght w Note that ths scheme the umber of relatos s w 1 has dfferet meag The dstace betwee two ectors s defed to be the half of the Hammg dstace betwee them Oe ca erfy that = ( w = p k j s ge by w k l=0 ( k l k w l w ( ( k w k w j l j l w Deote by T(w 1 1 w d the maxmum umber of bary ectors of legth 1 hag mutual Hammg dstace of at least d each ector has exactly w 1 oes the frst 1 coordates exactly w oes the last coordates By substtutg max{nc(hce1} T(e1we1 w4e (9 we obta the followg boud Theorem 1: A(4ew w w U w ( = max{a e1 } e1 w e1 ( e1 e U w( T(e1we1 w4e subject to Delsarte s lear costrats for Johso scheme (see [10 Theorem 1 p 666] Applyg Theorem 1 for e = 1 we obta the followg mproemets (the alues the paretheses are the best bouds preously kow [1] [14]: A( (50 A( (5064 A( (4080 We would lke to remark that LP ca be appled for upper bouds that obtaed by ceterg a spheres aroud a codewords We ge a example of such boud Theorem 13: A(10w ( ( w w 3 T(3w3 w10 3 w 4 w 4 T(4w4 w10 U w( 5 U w ( = max{ T(4w4 w10 A e 100 T(3w3 w T(4w4 w10 A e1 } subject to Delsarte s lear costrats for Johso scheme By Theorem 13 we hae: A( (81 A( (119 A( (158 A( (99 A( (81 ACKNOWLEDGMENT The work of Beam Mouts was supported part by grat o 63/04 of the Israel Scece Foudato The work of Tu Etzo was supported part by grat o 63/04 of the Israel Scece Foudato The work of Smo Ltsy was supported part by grat o 533/03 of the Israel Scece Foudato REFERENCES [1] E Agrell A Vardy K Zeger Upper bouds for costat-weght codes IEEE Tras o Iform Theory ol 46 pp No 000 [] E Agrell A Vardy K Zeger A table of upper bouds for bary codes IEEE Tras o Iform Theory ol 47 o 7 pp No 001 [3] M R Best Bary codes wth a mmum dstace of four IEEE Tras o Iform Theory ol 6 pp No 1980 [4] M R Best A E Brouwer The Trply Shorteed Bary Hammg Code Is Optmal Dscrete Mathematcs ol 17 pp [5] M R Best A E Brouwer F J MacWllams A M Odlyzko N J A Sloae Bouds for bary codes of legth less tha 5 IEEE Tras o Iform Theory ol 4 pp Ja 1978 [6] Ph Delsarte A algebrac approach to the assocato schemes of codg theory Phlps Research Reports Supplemets No [7] I Hokala Bouds for bary costat weght coerg codes Lcetate thess Departmet of Mathematcs U of Turku Turku Fl Mar 1987 [8] S M Johso A ew upper boud for error-correctg codes IRE Tras o Iform Theory ol 8 pp [9] S Ltsy A updated table of the best bary codes kow Hbook of Codg Theory (V S Pless W C Huffma eds ol 1 pp Amsterdam: Elseer 1998 [10] F J MacWllams N J A Sloae The Theory of Error-Correctg Codes Amsterdam: North-Holl 1977 [11] B Mouts T Etzo S Ltsy Improed Upper Bouds o Szes of Codes IEEE Tras o Iform Theory ol 48 pp Aprl 00 [1] C L N a Pul O bouds o codes Master s thess Dept of Mathematcs Computg Scece Edhoe U of Techology Edhoe the Netherls Aug 198 [13] C Roos C de Vroedt Upper Bouds for A(4 A(6 Dered from Delsarte s Lear Programmg Boud Dscrete Mathematcs ol 40 pp [14] A Schrjer New code upper bouds from the Terwllger algebra preprt Apr 004