Weight Distributions of Hamming Codes (II)

1 W eight Distrib utions of Hamming Cod es (II) Dae San Kim, Member , I EEE Abstract — In a previous pap er , we derive d a recursiv e formula determining the weight distributions of th e [ n = ( q m − 1) / ( q − 1) , n − m, 3] Hamming code H ( m, q ) , when ( m, q − 1) = 1 . Here q i s a prime power . W e note h ere that t he formula actually holds fo r any positive integer m and any prime p ower q , without the restriction ( m, q − 1) = 1 . Index T erms — Hamming code, weight distribution, Pless power moment identity . I . I N T R O D U C T I O N The q -ary Ham ming code H ( m, q ) is an [ n = ( q m − 1) / ( q − 1) , n − m, 3 ] code which is a sin gle-erro r-correcting perfect code. From now on, q will ind icate a p rime power un less otherwise stated . Also, assume that m > 1 . Moisio discovered a hand ful of n ew power m oments of Kloosterman sums over F q , when the characteristic of F q is 2 and 3 ([3], [4], [ 6], [7]). The idea is, v ia Pless power moment id entity , to con nect mome nts of Kloosterm an sums and frequen cies of weights in the binary Zetterberg code of length q + 1 , or those in th e ternary Me las code o f len gth q − 1 . In [1], we adopted his idea of u tilizing Pless power mo ment identity and exponential sum techniques so that we were able to derive The orem 1 below u nder the restriction that ( m, q − 1) = 1 . Th is restriction was n eeded to assume that H ( m, q ) is cyclic (cf. Th eorem 3). It is somewhat surprising that there has been no suc h recursive formu las g iving the weight distrib utio ns of th e Ham ming codes in the non binary cases, whereas th ere has been o ne in the binary case(cf . The orem 2). In this co rrespon dence, we will g iv e an elementar y pr oof showing that th e restriction ( m, q − 1) = 1 can b e r emoved. Theor em 1: Let { C h } n h =0 ( n = ( q m − 1) / ( q − 1)) d enote the weight distribution o f the q - ary Hamming co de H ( m, q ) . Then, fo r h with 1 ≤ h ≤ n , h ! C h =( − 1) h q m ( h − 1) ( q m − 1) + h − 1 X i =0 ( − 1) h + i +1 C i h X t = i t ! S ( h, t ) q h − t ( q − 1) t − i ( n − i n − t ) , (1) where S ( h, t ) denotes the Stirling num ber of th e second kin d defined by S ( h, t ) = 1 t ! t X j =0 ( − 1) t − j ( t j ) j h . (2) Theor em 2 (p.1 29 in [2 ]): L et { C h } n h =0 ( n = (2 m − 1)) denote the weight d istribution o f the bina ry Hamming code This work was supported by grant No. R01-2007-000-11 176-0 from the Basic Research Program of the Korea Science and Engineering Foundation. The author is with the Department of Mathematics, Sogang Univ ersity , Seoul 121-742, Korea(e -mail: dskim@sogang.ac.kr). H ( m, 2) . Then the weigh t d istribution satisfies the fo llowing recurren ce relation: C 0 = 1 , C 1 = 0 , ( i + 1) C i +1 + C i + ( n − i + 1) C i − 1 = ( n i ) ( i ≥ 1) . Theor em 3 ([5]): Let n = ( q m − 1 ) / ( q − 1) , whe re ( m, q − 1) = 1 . Let γ b e a prim iti ve ele ment of F q m . Then the cyclic code of length n with the defining zero γ q − 1 is eq uiv alent to the q -ary Ha mming co de H ( m, q ) . I I . P R O O F O F T H E O R E M 1 W e know that th e for mula (1) h olds for ( m, q − 1) = 1 ( [1, Theorem 1]) . By the r ecursive fo rmula in (1), we see that all C i ( i = 0 , 1 , 2 , · · · , n = ( q m − 1) / ( q − 1)) are form ally polyno mials in q with ration al coefficients, wh ich depe nd on m (cf. Corollary 2 in [1] f or the explicit expressions of C i for i ≤ 10) . Put C i = P i ( q ; m ) , fo r i = 0 , 1 , 2 , · · · , n = ( q m − 1) / ( q − 1) . T hen ( 1) can be rewritten as h ! P h ( q ; m ) = ( − 1) h q m ( h − 1) ( q m − 1)+ h − 1 X i =0 ( − 1) h + i +1 P i ( q ; m ) h X t = i t ! S ( h, t ) q h − t ( q − 1) t − i  q m − 1 q − 1 − i t − i  , (3) (1 ≤ h ≤ n = ( q m − 1) / ( q − 1)) . Let m , h be fixed positive integers. Then the LH S an d the RHS of (3) a re formally poly nomials in q and (3) is valid whenever q is replaced by prime powers p r satisfying ( m, p r − 1) = 1 and h ≤ ( p r m − 1) / ( p r − 1) . So it is eno ugh to show th at there are infin itely many p rime powers p r such that ( m, p r − 1) = 1 , since then (3) is really a polyno mial identity in q , so that the restriction of our concern can b e removed. There are thr ee cases to be considered. Case 1) 2 does n ot divide m . Let m = p e 1 1 p e 2 2 · · · p e r r , where p 1 , p 2 , · · · , p r are d istinct odd primes an d e j ’ s are p ositi ve integers. T hen, by Dirich let’ s theorem on arithmetic progre ssions, there are infinitely many prime numb ers p such tha t p ≡ 2 (mod m ). For each such an p , p ≡ 2 (mod p j ), for j = 1 , · · · , r . Then p j does not divide p − 1 , for all j , so that all p j is relatively prime to p − 1 . So ( m, p − 1) = 1 , f or all such primes p . Case 2) 2 is the on ly pr ime divisor of m . In this case, 2 l − 1 ( l = 1 , 2 , · · · ) are all relativ ely prime to m . Case 3) 2 and some odd prime divide m . Let m = 2 e m 1 , m 1 = p e 1 1 p e 2 2 · · · p e r r , where e, e 1 , · · · , e r , r are positi ve integers and p 1 , p 2 , · · · , p r are distinct odd primes. Noting that (2 , m 1 ) = 1 , we let f = ord m 1 2 be th e order of 2 mod ulo m 1 . Then 2 lf ≡ 1 (m od m 1 ), for all positive integers l . So 2 lf ≡ 1 (mo d p j ), fo r all j = 1 , · · · , r . Thus 2 lf +1 ≡ 2 (mod p j ), for all j , and hence p j does not divide 2 2 lf +1 − 1 , for all j . This implies that ( m, 2 lf +1 − 1 ) = 1 , for all positi ve integers l .  R E F E R E N C E S [1] D. S. Kim, “W eight distribut ions of Hamming codes, ” submitted. [2] F . J. MacW illiams and N. J. A. Sloane, The Theory of Error Correctin g Codes. Amsterdam, The Netherlands : North-Holla nd, 1998 . [3] M. Moisio, “The moments of a Kloosterman sum and the weight distrib ution of a Zetterber g type binary cy clic code, ” IEEE T rans. Inf. Theory , vol. IT -53, pp. 843-847, 2007. [4] M. Moisio, “On the moments of Klooste rm an sums and fibre products of Kloosterman curves, ” Fi nite Fiel ds Appl., in Press. [5] V . S. Pless, W . C. Huf fman, and R. A. Brualdi, “ An introduc tion to algebra ic codes, ” in Handbook of Coding Theroy , V . S. P less and W . C. Huf fman, Eds. Amsterdam, The Netherla nds : North-Ho lland, 1998, vol. I, pp. 3-139. [6] R. Schoof and M. van der Vlugt, “Heck e operators and the weigh t distrib ution of certai n codes, ” J. Combin. Thero y Ser . A, vol. 57, pp. 163-186, 1991. [7] G. v an der Geer , R. Schoof and M. van der Vlugt , “W eig ht formulas for ternary Melas codes, ” Math. Comp., vol 58, pp. 781-792, 1992.