Jurnal Teknologi THE EQUIVALENT IDENTITIES OF THE MACWILLIAMS IDENTITIES FOR LINEAR CODES. Full Paper. Bao Xiaomin *

Transcription

1 Junal Teknolog THE EQUIVALENT IDENTITIES OF THE MACWILLIAMS IDENTITIES FOR LINEAR CODES Bao Xaomn * School of Mathematcs & Statstcs, Southwest Unvest, Chongng, , Chna Full Pape Atcle hsto Receved 29 Janua 2015 Receved n evsed fom 24 Mach 2015 Accepted 1 August 2015 *Coespondng autho xbao@swueducn Abstact We use devatves to pove the euvalences between MacWllams dentt and ts fou euvalent foms, and pesent new ntepetatons fo the fou euvalent foms Kewods: Lnea code, Hammng weght, MacWllams dentt, euvalent, devatve 2015 Penebt UTM Pess All ghts eseved 10 INTRODUCTION Let be a (n, k) lnea code on the feld F GF ( ) and let be ts dual code Defne: and The followng dentt s poved b [6] and s called the MacWllams dentt: The followng ae fou euvalent foms of the MacWllams dentt: The MacWllams dentt and the fou euvalent foms have been studed b man authos [1-6, 8, 9] In 1963, MacWllams [6] poved that (2), (3) and (4) ae all euvalent to MacWllams dentt (1) In 1983, b usng a method dffeent fom that of [6], Blahut [1] poved that (1) can be deved fom (4) Smla method can also be used to deve (1) fom (3) Identt (5) was ntall dscoveed b Buald et al n 1980 [2], and the showed that (5) can be deved fom (2) In 1997, Goldwasse [4] poved (5) b nducton It should be ponted out that Buald et al pesented combnatoal ntepetatons fo (3), (4) and (5) n [2] Let M be a n matx whose ows ae the codewods of n some ode Let be an ntege wth 0 n A ow of M wth weght contans -tuples of nonzeos So the numbe of -tuples of nonzeo n n the ows of M euals W Hence dentt (3) s a 0 conseuence of countng the numbe of -tuples of nonzeos n the ows of M n two dffeent was 76:1 (2015) wwwunalteknologutmm eissn

2 382 Bao Xaomn / Junal Teknolog (Scences & Engneeng) 76:1 (2015) Smlal, dentt (4) s a conseuence of countng the numbe of -tuples of zeos n the ows M n two dffeent was; whle dentt (5) s a conseuence of countng the numbe of -tuples of weght t n the ows of M n two dffeent was Accodng to the ntepetatons, both (3) and (4) ae specal cases of (5) In the followng secton we wll use devatves to pove the euvalence between anone of (2), (3), (4), (5) and (1), ou poofs also unvel new elatonshps between MacWllams dentt and ts euvalent foms 20 PROOFS OF EQUIVALENCES The followng two lemmas ae needed n ou euvalence poofs: Lemma 1 Let X x, Y x, f X s Y t, then fo an non-negatve nteges l, m we have the othe two cases can be poved smlal Let Lemma 2 Let f ( x, ) and g( x, ) be two homogeneous polnomals of degee n n x, If then fom (6) we can get the followng euatons: then f ( x, ) = g ( x, ) Poof of Lemma 1 We onl pove the second dentt, the fst one can be poved smlal If m = 0, the esult s obvous Now let m>0, and suppose Solvng these euatons we get fn gn, fn1gn1, º, f1g1, f0 g0 Theefoe f ( x, ) g ( x, ) 21 Deve (2) o (3) fom (1) B takng -th patal devatve wth espect to on both sdes of (1), we get The asseton follows b nducton Poof of Lemma 2 We onl pove the case of

3 383 Bao Xaomn / Junal Teknolog (Scences & Engneeng) 76:1 (2015) Substtutng 1 fo x, 0 fo n the above euaton we get and lemma 1 we get So fom (1) we can deve (2) - Substtutng 1 fo both x and we get Theefoe, fom (1) we can deve (3) 22 Deve (4) fom (1) B takng -th patal devatve wth espect to x on both sdes of (1), we get Substtung 1 fo both x and, and also notce that when we get ( x - ) -- s 0 s n n 1 W W 0 t t n k 0 t n n t ( 1) t n 0 t t k t n ( 1) t W 0 0 t So (5) holds 24 Deve (1) fom (2) So fom (1) we can deve (4) 23 Deve (5) fom (1) Let n f ( x, ) W ( x, ) W x n 0 1 g ( x, ) W ( x ( 1), x ) nk 1 n W [ x ] n ( x ) nk 0 Then both f(x,) and g(x,) ae homogeneous polnomals of degee n n x, Fo an non-negatve ntege, b lemma 1 we have n Let f ( x, ) W ( x, ) Fo 0t n, b takng -th mxed patal devatves on both sdes of we can get f t n n t ( t! W x ) x t t x t 0 t n n n t t t!( t )! W x 0 t t Fom

4 384 Bao Xaomn / Junal Teknolog (Scences & Engneeng) 76:1 (2015) f! W x1, 0 g 1 n! n W 1, 0 nk 0 0 x [ x ] n ( x ) x1, 0 1 n! W n nk 0 0 Snce (2) holds, we get B lemma 2 we obtan 30 CONCLUSION A homogeneous polnomal of degee n n two vaables s unuel detemned b ts n+1 coeffcents, and an popel selected n+1 ponts on the ange of the polnomal can be used to detemned these coeffcents Fom the poofs n the last secton we see that denttes (2), (3), (4) and (5) ae actuall fou dffeent goups of condtons that can be used to detemne the coeffcents of (1), and the can be wtten espectvel n the followng fou foms: 25 Deve (1) fom (3) o (4) We onl pove that fom (3) we can deve (1) Let Then both f(x,) and g(x,) ae homogeneous polnomals of degee n n x, Fo an non-negatve ntege n, b lemma 1 we get f n! W 1, 1 0 x g 1 n! n W n k 1, x Fom (3) we get [ x ] n ( x ) x1, 1 k n! n W 0 26 Deve (1) fom (5) If t = 0, then (5) educes to (4), whle f t =, then (5) educes to (3) Snce (1) can be deved fom (3) o (4), (1) can also be deved fom (5) Theefoe moe euvalent foms of (1) can be wtten out n ths wa Refeences [1] Blahut R E 1984 Theo and Pactce of Eo Contol Codes Readngs, Mass: Addson Wesle [2] Buald R A, V S Pless and J Bessnge 1988 On the MacWllams Identtes fo Lnea Codes Lnea Algeba Appl 107: [3] Chang, S C and J K Wolf 1980 A Smple Devaton of the MacWllams' Identt fo Lnea Codes IEEE Tan On Infom Theo IT-26(4): [4] Goldwasse, J L 1997 Shotened and Punctued Codes and the MacWllams Identtes Lnea Algeba Appl 253: 1-13 [5] Honold, T 1996 A Poof of MacWllams' Identt J of Geomet 57: [6] MacWllams, F J 1963 A Theoem on the Dstbuton of Weghts n a Sstematc Code Bell Sstem Tech J 42: [7] MacWllams, F J and N J A Sloane 1977 The Theo of Eo-Coectng Codes New Yok: Noth-Holland Publshng Compan

5 385 Bao Xaomn / Junal Teknolog (Scences & Engneeng) 76:1 (2015) [8] Pless, V S 1989 Intoducton to the Theo of Eo- Coectng Codes 2 nd ed New Yok:Wle-Intescence [9] Zele, N 1973 On the MacWllams Identt J Combnatoal Theo (A) 15: