Abstract
Using double counting, we prove Delsarte inequalities for \(q\)-ary codes and their improvements. Applying the same technique to \(q\)-ary constant-weight codes, we obtain new inequalities for \(q\)-ary constant-weight codes.
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References
Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Repts. Suppl. no. 10 (1973)
MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977)
Kang B.G., Kim H.K., Toan P.T.: Delsarte’s linear programming bound for constant-weight codes. IEEE Trans. Inf. Theory 58(9), 5956—5962 (2012)
Östergård P.R.J., Svanström M.: Ternary constant weight codes. Electron. J. Combin. 9(1), 41 (2002)
Ashikhmin A., Simonis J.: On the Delsarte inequalities. Linear Algebra Appl. 269, 197–217 (1998)
Simonis J., de Vroedt C.: A simple proof of the Delsarte inequalities. Des. Codes Cryptogr. 1, 77–82 (1991)
Schrijver A.: New code upper bounds from the Terwilliger algebra and semidefinite programming. IEEE Trans. Inf. Theory 51(8), 2859–2866 (2005)
Kim H.K., Toan P.T.: Improved semidefinite programming bound on sizes of codes. IEEE Trans. Inf. Theory 59(1), 7337–7345 (2013)
Gijswijt D.C., Schrijver A., Tanaka H.: New upper bounds for nonbinary codes based on the Terwilliger algebra and semidefinite programming. J. Combin. Theory Ser. A 113(8), 1719–1731 (2006)
Gijswijt D.C., Mittelmann H.D., Schrijver A.: Semidefinite code bounds based on quadruple distances. IEEE Trans. Inf. Theory 58(5), 2697–2705 (2012)
Johnson S.M.: A new upper bound for error-correcting codes. RE Trans. Inform. Theory 8, 203–207 (1962)
Cary Huffman W., Vera Pless: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Mounits B., Etzion T., Litsyn S.: Improved upper bounds on sizes of codes. IEEE Trans. Inf. Theory 48(4), 880–886 (2002)
Acknowledgments
The authors would like to thank the reviewer for his/her suggestions and comments, which improve the presentation of the paper. The work of H. K. Kim was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) (2010-0026473 and 2013053914).
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.
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Kim, H.K., Toan, P.T. New inequalities for \(q\)-ary constant-weight codes. Des. Codes Cryptogr. 73, 369–381 (2014). https://doi.org/10.1007/s10623-014-9924-x
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DOI: https://doi.org/10.1007/s10623-014-9924-x