Undetected error probability of q-ary constant weight codes

In this paper, we introduce a new combinatorial invariant called q-binomial moment for q-ary constant weight codes. We derive a lower bound on the q-binomial moments and introduce a new combinatorial structure called generalized (s, t)-designs which could achieve the lower bounds. Moreover, we employ the q-binomial moments to study the undetected error probability of q-ary constant weight codes. A lower bound on the undetected error probability for q-ary constant weight codes is obtained. This lower bound extends and unifies the related results of Abdel-Ghaffar for q-ary codes and Xia-Fu-Ling for binary constant weight codes. Finally, some q-ary constant weight codes which achieve the lower bounds are found.

[1]  Torleiv Kløve,et al.  On the undetected error probability for binary codes , 2003, IEEE Trans. Inf. Theory.

[2]  Rossitza Dodunekova.  Extended Binomial Moments of a Linear Code and the Undetected Error Probability , 2003, Probl. Inf. Transm..

[3]  Jehoshua Bruck,et al.  MDS array codes for correcting a single criss-cross error , 2000, IEEE Trans. Inf. Theory.

[4]  Bella Bose,et al.  Design of Balanced and Constant Weight Codes for VLSI Systems , 1998, IEEE Trans. Computers.

[5]  Haim Hanani,et al.  On Some Tactical Configurations , 1963, Canadian Journal of Mathematics.

[6]  Khaled A. S. Abdel-Ghaffar,et al.  A lower bound on the undetected error probability and strictly optimal codes , 1997, IEEE Trans. Inf. Theory.

[7]  Yixian Yang,et al.  On the undetected error probability of nonlinear binary constant weight codes , 1994, IEEE Trans. Commun..

[8]  San Ling,et al.  A Lower Bound on the Probability of Undetected Error for Binary Constant Weight Codes , 2006, 2006 IEEE International Symposium on Information Theory.

[9]  Fan Chung Graham,et al.  Optical orthogonal codes: Design, analysis, and applications , 1989, IEEE Trans. Inf. Theory.

[10]  Alexander Barg,et al.  Binomial Moments of the Distance Distribution and the Probability of Undetected Error , 1999, Des. Codes Cryptogr..

[11]  Tuvi Etzion,et al.  Optimal constant weight codes over Zk and generalized designs , 1997, Discret. Math..

[12]  Alexander Barg,et al.  Binomial Moments of the Distance Distribution: Bounds and Applications , 1999, IEEE Trans. Inf. Theory.

[13]  T. Kløve,et al.  On the Undetected Error Probability of m-out-of-n Codes on the Binary Symmetric Channel , 2000 .

[14]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[15]  Torleiv Kløve,et al.  Error detecting codes - general theory and their application in feedback communication systems , 1995, The Kluwer international series in engineering and computer science.

[16]  Torleiv Kløve,et al.  The undetected error probability threshold of m-out-of-n codes , 2000, IEEE Trans. Inf. Theory.

[17]  Shu-Tao Xia,et al.  Binary Constant-Weight Codes for Error Detection , 1998, IEEE Trans. Inf. Theory.

[18]  Jean-Marie Goethals,et al.  On the Nonbinary Johnson Scheme , 1985, Eur. J. Comb..

[19]  Jehoshua Bruck,et al.  Coding for tolerance and detection of skew in parallel asynchronous communications , 2000, IEEE Trans. Inf. Theory.