Abstract
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.
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References
Abdel-Ghaffar K.A.S. (1997). A lower bound on the undetected error probability and strictly optimal codes. IEEE Trans. Inform. Theory 43, 1489–1502
Ashikhmin A., Barg A. (1999). Binomial moments of the distance distribution: bounds and applications. IEEE Trans. Inform. Theory 45, 438–452
Barg A., Ashikhmin A. (1999). Binomial moments of the distance distribution and the probability of undetected error. Desi. Codes and Cryptogr. 16, 103–116
Blaum M., Bruck J. (2000). Coding for tolerance and detection of skew in parallel asynchronous communications. IEEE Trans. Inform. Theory 46, 2329–2335
Chung F.R.K., Salehi J.A., Wei V.K. (1989). Optical orthogonal codes: Design, analysis, and applications. IEEE Trans. Inform. Theory 35, 595–604
Dodunekova R. (2003). Extended binomial moments of a linear code and undetected error probability. Probl. Inform. Transmission 39, 255–265
Etzion T. (1997). Optimal constant weight codes over Z k and generalized designs. Discrete Math. 169, 55–82
Fu F.-W., Xia S.-T. (1998). Binary constant weight codes for error detection. IEEE Trans. Inform. Theory 44, 1294–1299
Fu F.-W., Kløve T., Wei V.K. (2003). On the undetected error probability for binary codes. IEEE Trans. Inform. Theory 49, 382–390
Fu F.-W., Kløve T., Xia S.-T.: On the undetected error probability of m-out-of-n codes on the binary symmetric channel. In: Buchmann J., Høholdt T., Stichtenoth H., Tapia-Recillas H. (eds.) Coding Theory, Cryptography, and Related Areas, pp. 102–110, Springer (2000).
Fu F.-W., Kløve T., Xia S.-T. (2000). The undetected error probability threshold of m-out-of-n codes. IEEE Trans. Inform. Theory 46, 1597–1599
Hanani H. (1963). On some tactical configurations. Canad. J. Math. 15, 702–722
Kløve T., Korzhik V. (1995). Error Detecting Codes: General Theory and Their Application in Feedback Communication Systems. Kluwer Acad. Press, Boston
MacWilliams F.J., Sloane N.J.A. (1981). The Theory of Error-Correcting Codes. North-Holland, Amsterdam
Mills W.H.: On the covering of triples by quadruple. In: Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Algorithms, pp. 573–581 (1974).
Tallini L.G., Bose B. (1998). Design of balanced and constant weight codes for VLSI systems. IEEE Trans. Comput. 47, 556–572
Tarnanen H., Aaltonen M., Goethals J-.M. (1985). On the nonbinary Johnson scheme. Eur. J. Combin. 6, 279–285
Wang X.M., Yang Y.X. (1994). On the undetected error probability of nonlinear binary constant weight codes. IEEE Trans. Commun. 42, 2390–2393
Xia S.-T., Fu F.-W., Jiang Y., Ling S. (2005). The probability of undetected error for binary constant weight codes. IEEE Trans. Inform. Theory 51, 3364–3373
Xia S.-T., Fu F.-W., Ling S. (2006). A lower bound on the probability of undetected error for binary constant weight codes. IEEE Trans. Inform. Theory 52, 4235–4243
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Xia, ST., Fu, FW. Undetected error probability of q-ary constant weight codes. Des. Codes Cryptogr. 48, 125–140 (2008). https://doi.org/10.1007/s10623-007-9130-1
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DOI: https://doi.org/10.1007/s10623-007-9130-1
Keywords
- Codes
- Constant weight codes
- Distance distribution
- Error detection
- Undetected error probability
- Generalized t-design