Derives an estimate for the error term in the binomial approximation of spectra of BCH codes. This estimate asymptotically improves on the bounds by Sidelnikov (1971), Kasami et al. (1985), and Sole (1990)

Krasikov, I

Litsyn, S

English

We derive an estimate for the error term in the binomial approximation of spectra of BCH codes. This estimate asymptotically improves on the earlier bounds by Sidelnikov, Kasami-Fujiwara-Lin and Solé

Ilia Krasikov

Simon Litsyn

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On Spectra of BCH Codes

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786 On Spectra of BCH Codes Ilia Krasikov and Simon Litsyn Abstruct- We derive an estimate for the error term in the binomial approximation of spectra of BCH codes. This estimate asymptotically improves on the earlier hounds by Sidelnikov, Kasami-Fujiwara-Lin, and Sult. Index Terms-BCH codes, spectra of codes, Krawtchouk polynomials. 1. INTRODUCTION I t  is well known that the distance distribution of binary primitive BCH codes can be approximated by the binomial distribution. If C is the BCH code of length i t  = 2"' - 1 and with minimum distance 2t  + 1 5 2'"''+1)'21 + 1. the components of the distance distribution vector I )  = ( [ I O . .  . . . b , ? )  can be written as b -A (") ( l + E , )  ' - ( ' I  + 1)' where \E,\  decreases with 1 ) .  Several upper bounds on \E, \  have been obtained. Sidelnikov [IS] gave a bound of order T I - "  for i in the relevant range. Further improvements on this estimate have been derived by Kasami, Fujiwara, and Lin [3] and Sol6 [16]. The bound of [3] for 1 1  large enough, t = o( fi) and i linearly depending on n ,  i = U J I ,  gives 1 - log, and the bound of 161 implies the following inequality: where H i m )  = -mlog2 ( U )  - (1 - crjlog, (1 - a). In this correspondence we present new estimates of the relative error term improving the known results. Our approach is based on investigating upper bounds for absolute values of Krawtchouk polynomials. Actually, this is just a refinement of an idea due to P. Sol6 [ 161. The bounds obtained are quite tight. Particularly, for t fixed our estimate is of order at most O (  i t - '  ). This approach can be easily generalized to arbitrary codes with known dual distance width, see e.g., [3]. As it was mentioned by an anonymous referee, the approach works as well for wide classes of Goppa codes, Melas codes, etc., where bounds of Carlitz-Uchiyama type for the dual distance width are known (see, e.g. [ 5 ]  ,161, [ I l l ,  [12]). Another upper estimates for values of Krawtchouk polynomials can be found in [4] where the expression for the envelope of absolute values of Krawtchouk polynomials was obtained. Manuscript received March 16, 1994; revised October 2, 1994. The work of  S. Litsyn was partly supported by the GUaStdllO Fellowship. I .  Krasikov is with Tel-Aviv University, School of Mathematical Sciences, Ramat-Aviv 69978 Tel-Aviv, Israel, and with Beit-Berl College, Kfar-Sava, Israel . S. Litsyn is with Tel-Aviv University, Department of Electrical Engineer- ing-Systems, Ramat-Aviv 69978 Tel-Aviv, Israel. IEEE Log Number 9410123. IEEE TRANSACTIONS ON INFORMATION THEORY. VOL. 41. NO. 3. MAY 1995 11. THE MAIN RESULT We will need some well known results about Krawtchouk polyno- The binary Krawtchouk polynomial PL' ( . r )  (of degree X .  in .r) is mials (see, e.g., [7], [SI, [ IO] .  [171). defined by the following generating function: x ~ P ; ~ ( . r ) ?  = (1 - : ) ) ( 1  + : ) ' l - . l .  (4) 1. =o Usually i t  is fixed and, when i t  does not lead to confusion, is omitted. We need some particular values of PA ( i ) ,  namely, From Cauchy's integral formula we get for nonnegative integer .I' (see, e.g., [16]): Thus for I I  and .I' even we have Hence, the following inequality holds (note, that in the sequel i is an integer, not J-1). Lemma 1: For i t  and i even, l P ~ ( i ) l  5 IPt ,p( i ) l .  Let us find the values P , 8 / 2 ( i )  for even 1 1  and i ' s .  The following symmetry relation holds for integer k and i (see, e.g. [SI): and So, from ( 5 )  and Lemma 1 we get for even i f  and i Employing (6) we obtain the following lemma. 0018-9448/95$04.00 0 1995 IEEE Authorized licensed use limited to: Brunel University. Downloaded on May 22, 2009 at 04:07 from IEEE Xplore.  Restrictions apply.IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 3, MAY 1995 181 Lemma 2: For k integer, and n and i even Theorem 1: In the extended BCH code of length n = 2m and minimum distance 2t + 2 5 2["'+']/' + 2  B, = 0, for i odd (" Now we are in a position to analyze the weight distribution of BCH codes. Consider the extended BCH code C of length n = 2", cardinality 2n-mt-1 , and minimum distance d = 2t + 2 5 2[("'+1)/21 + 2. Let the distance distribution of the code be B = (9) (Bo . . . . . 4 , ) , 40=B,=1 ,B21+1  = O f o r 1 = 0  ,.... n / 2 - 1 ,  and B,  = B,,-, = 0 for i = 1 , 2 , .  . . .2 t+1.  For usual (nonextended) BCH code of length n - 1 and minimum distance 2t + 1 we denote we obtan the following result for usual BCH its spectrum by b = ( b o , .  . . . b, - l ) .  Since the extended BCH code is doubly transitive we have the following result relating the values Theorem 2: In the BCH code of length = - = 2m - and of odd and even components of the spectra of BCH codes (see, e.g., [ lo ,  ch. 8.5, Theorems 14 and 161). B, = -+(l + E z ) ,  for 1 even n where nt L 7 2 )  ($) (:)(io ' IEzl I Using Lemma minimum distance 2t + 2[(m+1)/2] + Lemma 3: (7) and B: = B:l-L for i = O . . . . .  u,  BA = Bk = 1, B;,+, = 0 for i = 0,.  . . . n / 2  - 1, and for 2t + 1 5 2[(mf')/21 + 1 merefore,  for t = o(n  + 1, (,,1;2) (3 I?: = Bi,-z = 0, f o r i  = l:...d' = [ n / 2  - ( t  - 1 ) d .  (8) In -= 2 ( t  - 1)' + o( l ) ,  The last relation is due to Weil-Carlitz-Uchiyama (see [ lo ,  ch. 9.91 and [9], [13], [14] for further refinements). Denote by D' the segment [d' ,  . . . . n - d' ] .  Note that for the considered range of t and for t = o ( 6 )  n n: = IC'I = 2n' .  1=0 Inverting (7) we have This leads to straightforward corollaries from Theorems 1 and 2. Corollary I :  If t = o( f i ) ,  and i grows linearly with n, i / n  = (r + o (  l ) ,  then 1 1 - log2 I E m  I - - - H ( o )  + o(1). 2 Comparing the result of the corollary with (2 )  and (3) we conclude that our estimate is better for all (r E ( 0 , l ) .  Assuming t = o ( n a ) ,  we give somehow sharper estimates, particularly good for small i ' s .  We use that for i = o ( 6 )  from Stirling approximation we have We consider only even i ' s ,  since, otherwise, B,  = 0. Hence i.e., for i even we have (1 + 4 1 ) ) .  Corollary2: If t = o ( n a ) ,  i = o(f i ) ,  then If t and i are constants then we have IEzI = O(nt-'l2) (it was proven also in [3]). Since the maximum error occurs in (EZt+21r  we Now since 1 1  and i are even we may use Lemma 2, thus getting always have (E,I = O ( n - ' ) .  Authorized licensed use limited to: Brunel University. Downloaded on May 22, 2009 at 04:07 from IEEE Xplore.  Restrictions apply.788 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 3, MAY 1995 Finally, for BCH codes we get Theorem 3: Let t = o ( n a )  and 1 = [ ( i  + 1) /2 ] ,  then in the BCH On Primitive BCH Codes with Unequal Error Protection Capabilities Robert H. Morelos-Zaragoza, Member ,  IEEE, code of length rt  = 2”‘ - 1 and with minimum distance 2t + 1 and Shu Lin, Fellow, IEEE where the error term is upperbounded as follows: Abstract-We present a class of binary primitive BCH codes that have unequal-error-protection (UEP) capabilities. We use a recent result on the span of their minimum weight vectors to show that binary primitive BCH codes, containing second-order punctured Reed-Muller (RM) codes of the same minimum distance, are binary-cyclic UEP codes. The values of the error correction levels for this class of binary LUEP codes are estimated. I (  rt  - 21) 7 1  r l ‘ - i (  + O (  ) ). Note: Zndex Terms-Unequal error protection codes, binary primitive BCH After the correspondence had been submitted we were informed that a similar, slightly weaker (by a factor fi), bound can be derived from arguments presented in [2]. Their approach is quite different from that of ours. ACKNOWLEDGMENT The authors are grateful to the anonymous referees for helpful suggestions. REFERENCES [ I  1 W. Feller, An Introduction to Probability T h e o n  and Its Applications. New York: Wiley, 1970. [Z] 1. Gashkov and V. Sidelnikov, “Linear ternary quasiperfect codes correcting double errors,” Prohl. Peredachi Inform., vol. 22, no. 4, pp. 4348,  1986. 131 T. Kasami, T. Fujiwara, and S. Lin, “An approximation to the weight distribution of binary linear codes,” IEEE Trans. Inform. Theory, vol. IT-31, no. 6, pp. 769-780, 1985. 14) I .  Krdsikov and S. Litsyn, “Bounds for Krawtchouk polynomials,” in preparation. 151 G. Lachaud, “Distribution of the weights of the dual code of the Melas code,” Discrete Math., vol. 79. pp. 103-106, 1989. 161 G. Lachaud and J. Wolfmann, “The weights of the orthogonals of the extended quadratic binary Goppa codes,” IEEE Trans. Inform. Theory, vol. 36, pp. 686-692, 1990. 171 V. Levenshtein, “Krawtchouk polynomials and universal bounds for codes and designs in Hamming spaces,’’ submitted for publication. 181 J.  H. van Lint, Introduction to Coding Theon.  New York: Springer- Verlag, 1992. 191 S. Litsyn, C. J. Moreno, and 0. Moreno, “Divisibility properties and new bounds for cyclic codes and exponential sums in one and several variables,” Applicable Algrbrci in Eng., Commun. Comput., vol. 5 ,  pp. 105-1 16, 1994. I 101 F. J .  MacWilliams and N. J .  A. Sloane, The T h e o n  ofError-Correcting Codes. New York: North-Holland, 1977. [ I  I ]  C. J .  Moreno and 0. Moreno, “Exponential sums and Goppa codes I,” Proc. Amer. Math. Soc., vol. I 1  I ,  pp. 523-53 I ,  1991. [ 121 -, “Exponential sums and Goppa codes 11,” IEEE Trans. Inform. Theon,  vol. 38, pp. 1222-1229, 1992. 1131 0. Moreno and C. J.  Moreno. “The MacWilliamr-Sloane conjecture on the tightness of the Carlitz-Uchiyama bound and the weights of duals of BCH codes,” IEEE Trans. Irform. Theon,  vol. 40, pp. 1894-1907, Nov. 1994. 1141 F. Rodier, “On the spectra of the duals of binary BCH codes of designed distance = 9,” IEEE Tram\. Inform. Theon,  vol. 38, pp. 418479, 1992. [ 151 V. M. Sidelnikov, “Weight spectrum of binary Bose-Chaudhuri- Hocquenghem codes,” Prohl. Peredachi Inform., vol. 7, no. I ,  pp 14-22, 1971. [ 161 P. Sole, ”A limit law on the distance distribution of binary codes,” IEEE Truns. Inform. Theon,  vol. 36, pp. 229-232, 1990. [ 171 G. Szego, Orthogonal Polynomicils. Providence, RI: Amer. Math. Soc. Colloq. Publ., vol. 23, 1975. I.  INTRODUCTION Unequal error protection codes protect some of the encoded message symbols against more errors than the error correction level given by their minimum Hamming distance. Linear unequal error protection (LUEP) codes were first introduced by Masnick and Wolf [I] .  They discussed linear codes, specified by their parity check matrices, providing a level of error correction beyond that given by the minimum distance of the code, for some codeword positions. Gore and Kilgus [2] introduced an example ( 1.5.9) binary-cyclic UEP code with minimum distance 4 that can correct one information bit against the occurrence of two errors. That is, the most significant bit can always be decoded in the presence of up to two random errors in a received vector. Since then, other cyclic UEP codes have been introduced [ 3 ] ,  [4]. Binary BCH codes form a popular family of cyclic codes that have found numerous practical applications, due to their ability to correct multiple random errors, as well as their efficient coding and decoding procedures. Therefore, i t  is of interest to find conditions under which binary BCH codes are binary LUEP codes. To analyze the multilevel error correcting capabilities of binary linear codes, the concept of set of minimum weight vectors is fundamental. Dejniriori I S ] :  Let C‘ be an ( t i .  k . d )  linear code. The set of minimum-weight codewords, denoted .M, is defined as where w t ( c )  denotes the Hamming weight of vector E ,  and t = With the above definition, Boyarinov and Katsman [SI found conditions for linear codes to be LUEP codes: Lemma I :  To provide the protection level 6 for at least k’ in- formation digits of an ( I / .  k .  d )  linear code C‘, i t  is necessary and L(d - 1) /2J .  Manuscript received August 18, 1993; revised May 2, 1994. Part of this work was supported by NASA under Grant NAG 5-931, and by the NSF under Grants NCR-88813480 and NCR-91154(W). The material in this correspondence was presented at the International Symposium on Information Theory, Trondheim, Norway, 1994. R. H. Morelos-Zaragoza was with the Department of Information and Com- puter Sciences, Faculty of Engineering Science, Osaka University, Toyonaka, Osaka 560, Japan. He is now with the Third Department, Institute of Industrial Science, University of Tokyo, 7-22- I ,  Roppongi, Minatoku, Tokyo 106, Japan. S. Lin is with the Department of Electrical Engineering, University of Hawaii at Manoa, Honolulu, HI 96822 USA. IEEE Log Number 94 104 17. 0018-9448/95$04.00 0 1995 IEEE Authorized licensed use limited to: Brunel University. Downloaded on May 22, 2009 at 04:07 from IEEE Xplore.  Restrictions apply.

A limit law on the distance distribution of binary codes,”

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An Introduction to Probability Theon and Its Applications.

at 04:07 from IEEE Xplore. Restrictions apply.

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Divisibility properties and new bounds for cyclic codes and exponential sums in one and several variables,”

Exponential sums and Goppa codes 11,”

Exponential sums and Goppa codes I,”

Introduction to Coding Theon.

Linear ternary quasiperfect codes correcting double errors,” Prohl.

revised

The T h e o n ofError-Correcting Codes.

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On spectra of BCH codes

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On spectra of BCH codes

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