The binomial distribution is a well-known approximation to the distance spectra of many classes of codes. We derive a lower estimate for the deviation from the binomial approximatio

Krasikov, I

Litsyn, S

English

I. Krasikov

S. Litsyn

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On the accuracy of the binomial approximation to the distance distribution of codes

Brunel University Research Archive

1412 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 5, SEPTEMBER 1995 [7] E. N. Gilbert, “A comparison of signaling alphabet,” Bell Syst. Tech. J . ,  vol. 31, pp. 504-522, 1952. IS ]  J. Kahn, G. Kalai, and N. Linial, “The influence of variables on Boolean functions,” in Proc. 29th Ann. Symp. on Foundations of Computer Science. 191 G. A. Kabatiansky and V. I. Levenshtein, “Bounds on packing on a sphere and in space,” Probl. Inform. Transmission, vol. 14, pp. 1-17, 1978. [ I O ]  R. J. McEliece and H. C. Rumsey, “Sphere-packing in the Hamming metric,” Bull. Amer. Math. Soc., vol. 75, pp. 32-34, 1969. [ 1 I ]  R. J. McEliece, E. R. Rodemich, H. C. Rumsey, and L. R. Welch, “New upper bounds on the rate of codes via the Delsarte-MacWilliams inequalities,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 157-166, 1977. [I21 F. J. MacWilliams and N. J. A. Sloane, The Theory ofError Correcting Codes. Amsterdam, The Netherlands: North-Holland, 1977. [I31 N. J. A. Sloane, “Recent bounds for codes, sphere packings and related problems obtained by linear programming and other methods,” Contemp. [ 141 J. H. van Lint, Introduction to Coding Theory. New York: Springer- Verlag, 1982. Los Alamitos, CA: Computer Soc. Press, 1988, pp. 68-80. Math., vol. 9, pp. 153-185, 1982. Lemma 1; Let F be the set of real monk  polynomials of degree e. Define Then provided U .  b > -112. real. of Jacobi polynomials (see, e.g., [ l o ] )  Proof: Let f be an optimal polynomial. Expand f in the series The leading coefficient of Pjn ”(s) is 2-J ( ‘r+’+2’ ), and so 2‘ q c  = ( 2 n  + 2h + 2 4  .On the Accuracy of the Binomial Approximation \ . I  The orthogonality relation for Jacobi polynomials is given by to the Distance Distribution of Codes Ilia Krasikov and Simon Litsyn 2“+j+1r(J + + i ) r (J+ + 1) 0,1 - - (2] + + 3 + i)r(J + im+ 0 + j + 1) Abstract-The binomial distribution is a well-known approximation to the distance spectra of many classes of codes. We derive a lower estimate for the deviation from the binomial approximation. where 15,l is the Kronecker delta. Index Terms-Spectra of codes, Krawtchouk polynomials. Now we get zE[ -1  I ]  niax (1 - . r ) “ ( ~  - .r)’(f(.r))’ I. INTRODUCTION The binomial distribution is a well-known approximation to the distance spectra of many classes of codes. For example, it is known to be tight for the weights of BCH codes (see, e.g. [7, sec. 9.101). Several upper bounds for the error term of such approximation have been derived in [ I ] ,  [2], [4], [8], [9]. These estimates show that, provided the dual distance is large enough, the spectrum of the code rapidly converges to the binomial distribution. How close can the real distribution be to the binomial one? In this correspondence we give a lower estimate for the deviation from the binomial approximation thus showing that it cannot be too sharp. We also establish an identity relating the error terms to the dual spectrum of a code. 11. RESULTS We start with the following auxiliary lemma [3]. The proof is presented for self-completeness. ’ P y a  L b ) ( , r . j  n.r 2 -qP gr< ( 2 n .  2 b )  1 2 and we are done. U The binary Krawtchouk polynomial PT(.r) (of degree k in .r) is defined by the following generating function: - p : ( . r ) . k  = ( l - r ) ” ( l + . ) ” - ” .  (1) h =O When it does not lead to confusion n is omitted, i.e., Pk(.r) = P t  (s). The following values are of importance for us: \ I  Manuscript received July 19, 1994; revised February 2, 1995. This research was partially supported by the Guastallo Fellowship and a Grant from the Israeli Ministry of Science and Technology. I. Krasikov is with Tel-Aviv University, School of Mathematical Sciences, Ramat-Aviv 69978, Tel-Aviv, Israel, and 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 94 13879. Let the distance distribution of a code c be B = (Bo.. . . . B7% 1, and - B’ = (BA.. . . . D:? ) stand for the the dual spectrum, that is, B‘ is determined by the ~ ~ ~ ~ i l l i ~ ~ ~  transform of 3 (2) 0018-9448/95$04.00 0 1995 IEEE Authorized licensed use limited to: Brunel University. Downloaded on May 22, 2009 at 04:20 from IEEE Xplore.  Restrictions apply.IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 41, NO. 5 ,  SEPTEMBER 1995 . B ~ , ( - l j J ( l - , l . ) ' - ~ ~ ( l + . r ) ~ ~ - '  ~ J = " ,  b1The inverse is given by , (3) Hence 71 - 1 B! = !wl+ 2 (-1)";) + D L P , ( k j .  L = l  Quite often the first term turns out to be dominating. Note also that DiL E [o. I] .  Define r ,  = B,  - __ ICl('l) (1 + (-1 ). 2" This is evidently the deviation of the rth spectrum element from the "expected" value given by the binomial distribution. Theorem 1. Let 13: = 0, for i E [l. d; - 11 U [d: + 1.11 - 11. Then Proofi Let / = a, 9 E [0,7r/2], and put .r = cos+.  Denote also 0 1  = [ ( d ;  + 1)/2] ,  bi = [d;/2], 0 2  = [d:/2], bz = [ ( d ;  - 1)/2] .  From the definition of r ,  and (3) Denote by E,'" and E,''' the sums over all even (odd) j E [d ; .  d ; ] .  Using (1) with t = p2'Iq, we  get 1473 and applying Lemma 1 with (/ = rr 1. b = i / / 2  - 61. c = bl - n I ,  to thetirsttermof(4),andwithcr = r r 2 + 1 / 2 ,  b =  (n -1 ) /2 -b2 .  c = 0 For wide classes of codes, d: = 71 - d ; .  For example, it is the case when the code contains only even weight vectors. For even n and (1: the estimate gets the form b2 - 0 2 .  to the second one, we get the result. Consider BCH codes of distance d = 2t + 1 < 6. Upper estimates for the distance of the code, obtained by extending the code dual to the BCH code, may be deduced from the lower bound on exponential sums (see, e.g. [ 5 ] )  (1: 5 11/2 - c 1 f i  for some constant c1. Then For constant f this estimate turns out to be asymptotically tight. This follows from results of [ I ] ,  [4] where it was shown that where H is the binary entropy function. In what follows we will derive an identity relating the deviations to the dual distance distribution. This is achieved by refining some arguments due to Gashkov and Sidelnikov [I] .  We need (see, e.g., [6]) the following properties of Krawtchouk polinomials (for integer i ,  j .  1, k E [O. ~ i ] ) :  Authorized licensed use limited to: Brunel University. Downloaded on May 22, 2009 at 04:20 from IEEE Xplore.  Restrictions apply.I474 Lemmci 2: IEEE TRANSACTIONS OK INFORMATION THEORY. VOL. 41, NO. 5.  SEPTEMBER 1995 Proof: Just follows from the evident d ; B: 5 2"/lCl - 1. A similar bound was obtained in [ l ]  by more complicated argu- ments. Theorem 2: Proof: Using (2) observe that r ,  = 0. , = ( I  and Hence 0 By the previous lemma this is also and we are done. 0 ACKNOWLEDGMENT The authors wish to thank V. Sidelnikov and P. Sol6 for helpful suggestions. REFERENCES [ 1 1 1. Gashkov and V. Sidelnikov, "Linear ternary quasiperfect codes correcting double errors," Probl. Peredachi Inform, vol. 22, no. 4, pp. 43-48, 1986. 121 T. Kasami, T. Fujiwara, and S. Lin, "An approximation to the weight distribution of binary linear codes," lEEE Trclns. Inforni. Theon, vol. IT-? I ,  no. 6, pp. 769-780, 1985. 131 I .  Krasikov, "Degree conditions for vertex switching reconstruction," submitted. [4] 1. Krasikov and S. Lityyn, "On spectra of BCH codes,'' lEEE Trcms. Inform. Thron ,  vol. 41, pp. 786-788, 1995. 151 V. Lcvcnshtein, "Bounds for packings of metric space5 and some their applications," in  Problem! Kibernetiki, vol. 40. Moscow, USSR: Nauka, 1983, pp. 43-110 (in Russian). 161 J. H. van Lint, lntroduc-rion to Coding Theon. New York: Springcr- Verlag, 1992. [7] F. J. MacWilliama and N. J .  A. Sloane, The Theory oJ'Error-Correcting Codes. New York: North-Holland, 1977. [ 8 ]  V. M. Sidelnikov, "Weight spectrum of binary Bosc-Chaudhuri- Hocquenghem codes,'' Probl. Pererlachi Inform., vol. 7 ,  no. I ,  pp. 14-22, 1971. 191 P. Sole. "A limit law on the distance distribution of binary codes," lEEE Trtln,. h@orm. Theory, vol. 36, pp. 229-232, 1990. [ 101 G. Szcgii, "Orthogonal polynomials," Amer. Math. Soc.. Colloq. Pub/., vol. 23. Providence, RI: Amer. Math. Soc., 1975. Authorized licensed use limited to: Brunel University. Downloaded on May 22, 2009 at 04:20 from IEEE Xplore.  Restrictions apply.

A limit law on the distance distribution of binary codes,&quot;

An approximation to the weight distribution of binary linear codes,&quot;

Bounds for packings of metric space5 and some their applications,&quot;

Degree conditions for vertex switching reconstruction,&quot; submitted. [4] 1. Krasikov

Linear ternary quasiperfect codes correcting double errors,&quot;

lntroduc-rion to Coding Theon.

On the accuracy of the binomial approximation to the distance distribution of codes

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