Abstract
Starting from a linear [n, k, d] q code with dual distance \({d^{\bot}}\) , we may construct an \({[n - d^\bot, k - d^\bot +1,\geq d]_q}\) code with dual distance at least \({\left\lceil\frac{d^\bot}{q}\right\rceil}\) using construction Y 1. The inverse construction gives a rule for the classification of all [n, k, d] q codes with dual distance \({d^{\bot}}\) by adding \({d^\bot}\) further columns to the parity check matrices of the smaller codes. Isomorph rejection is applied to guarantee a small search space for this iterative approach. Performing a complete search based on this observation, we are able to prove the nonexistence of linear codes for 16 open parameter sets [n, k, d] q , q = 2, 3, 4, 5, 7, 8. These results imply 217 new upper bounds in the known tables for the minimum distance of linear codes and establish the exact value in 109 cases.
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References
Barnabei M., Searby D., Zucchini C.: On small (k, q)-arcs in planes of order q 2. J. Combin. Theory Ser. A 24, 241–246 (1978)
Betten A., Braun M., Fripertinger H., Kerber A., Kohnert A., Wassermann A.: Error-Correcting Linear Codes. Springer, Berlin (2006)
Bosma W., Cannon J. J., Fieker C., Steel A. (eds.): Handbook of Magma Functions, Edition 2.16 (2010).
Bouyukliev I.G., Jacobsson E.: Results on binary linear codes with minimum distance 8 and 10. IEEE Trans. Inform. Theory 57, 6089–6093 (2011)
Edel Y., Bierbrauer J.: Inverting construction Y 1. IEEE Trans. Inform. Theory 44, 1993 (1998)
Feulner T.: The automorphism groups of linear codes and canonical representatives of their semilinear isometry classes. Adv. Math. Commun. 3, 363–383 (2009)
Feulner T., Nebe G.: The automorphism group of a self-dual binary [72, 36, 16] code does not contain Z 7, Z 3 × Z 3 or D 10. arXiv:abs/1110.6012 (2011).
Grassl M.: Bounds on the minimum distance of linear codes and quantum codes (2012). http://www.codetables.de. Accessed 22 Feb 2012.
Huffman W.C., Pless V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)
Schmid W., Schürer R.: MinT: A database for optimal net parameters. In: Niederreiter, H., Talay, D. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004, pp. 457–469, Springer, Heidelberg (2006). http://mint.sbg.ac.at/. Accessed 22 Feb 2012.
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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding Theory and Applications”.
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Feulner, T. Classification and nonexistence results for linear codes with prescribed minimum distances. Des. Codes Cryptogr. 70, 127–138 (2014). https://doi.org/10.1007/s10623-012-9700-8
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DOI: https://doi.org/10.1007/s10623-012-9700-8