Abstract
Codes from adjacency matrices from the Hamming graphs \(H^k(n,m)\) are examined for the property of being special LCD codes. The special property involves being able to propose a feasible decoding algorithm for the binary codes, and also to deduce the dimension of the code from the eigenvalues of an adjacency matrix, which are known for these graphs. Some positive results are obtained, in particular for the binary and ternary codes.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Assmus, E.F. Jr, Key, J.D.: Designs and Their Codes. Cambridge University Press, Cambridge (1992), Cambridge Tracts in Mathematics, vol. 103 (Second printing with corrections, 1993)
Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: the user language. J. Symb. Comput. 24, 3/4, 235–265 (1997)
Cannon, J., Steel, A., White, G.: Linear codes over finite fields. In: Cannon, J., Bosma, W. (eds.) Handbook of Magma Functions, Computational Algebra Group, Department of Mathematics, University of Sydney, V2.13, pp. 3951–4023 (2006). http://magma.maths.usyd.edu.au/magma
Fish, W.: Binary codes and partial permutation decoding sets from the Johnson graphs. Gr. Combin. 31, 1381–1396 (2015)
Fish, W., Fray, R., Mwambene, E.: Binary codes and partial permutation decoding sets from the odd graphs. Cent. Eur. J. Math. 12(9), 1362–1371 (2014)
Fish, W., Key, J.D., Mwambene, E.: Codes, designs and groups from the Hamming graphs. J. Combin. Inform. Syst. Sci. 34(1–4), 169–182 (2009)
Fish, W., Key, J.D., Mwambene, E.: Graphs, designs and codes related to the \(n\)-cube. Discrete Math. 309, 3255–3269 (2009)
Fish, W., Key, J.D., Mwambene, E.: Binary codes from designs from the reflexive n-cube. Util. Math. 85, 235–246 (2011)
Fish, W., Key, J.D., Mwambene, E.: Ternary codes from reflexive graphs on 3-sets. Appl. Algebra Eng. Commun. Comput. 25, 363–382 (2014)
Fish, W., Key, J.D., Mwambene, E.: Binary codes from reflexive graphs on \(3\)-sets. Adv. Math. Commun. 9, 211–232 (2015)
Ghinelli, D., Key, J.D., McDonough, T.P.: Hulls of codes from incidence matrices of connected regular graphs. Des. Codes Cryptogr. 70, 35–54 (2014). https://doi.org/10.1007/s10623-012-9635-0
Haemers, W.H., Peeters, R., van Rijckevorsel, J.M.: Binary codes of strongly regular graphs. Des. Codes Cryptogr. 17, 187–209 (1999)
Huffman, W.C.: Codes and groups. In: Pless, V.S., Huffman, W.C. (eds.) Handbook of Coding Theory, vol. 2, pp. 1345–1440. Elsevier, Amsterdam (1998). Part 2, Chapter 17,
Key, J.D., Moori, J., Rodrigues, B.G.: Binary codes from graphs on triples. Discrete Math. 282(1–3), 171–182 (2004)
Key, J.D., Moori, J., Rodrigues, B.G.: Permutation decoding for binary codes from triangular graphs. Eur. J. Combin. 25, 113–123 (2004)
Key, J.D., Moori, J., Rodrigues, B.G.: Ternary codes from graphs on triples. Discrete Math. 309, 4663–4681 (2009)
Key, J.D., Rodrigues, B.G.: Special \({LCD}\) codes from Peisert and generalized Peisert graphs. Graphs Combin. (2019). https://doi.org/10.1007/s00373-019-02019-0
Key, J.D., Rodrigues, B.G.: \({LCD}\) codes from adjacency matrices of graphs. Appl. Algebra Eng. Commun. Comput. 29(3), 227–244 (2018)
Key, J.D., Seneviratne, P.: Permutation decoding for binary self-dual codes from the graph \({Q}_n\) where \(n\) is even. In: Shaska, T., Huffman, W.C., Joyner, D., Ustimenko, V. (eds.) Advances in Coding Theory and Cryptology, pp. 152–159. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ (2007). Series on Coding Theory and Cryptology, 2
Key, J.D., Fish, W., Mwambene, E.: Codes from the incidence matrices and line graphs of Hamming graphs \({H}^k(n,2)\) for \(k \ge 2\). Adv. Math. Commun. 5, 373–394 (2011)
Leemans, D., Rodrigues, B.G.: Linear codes with complementary duals from some strongly regular subgraphs of the McLaughlin graph. Math. Commun. 21(2), 239–249 (2016)
Massey, J.L.: Linear codes with complementary duals. Discrete Math. 106(107), 337–342 (1992)
Mullin, N.: Self-complementary arc-transitive graphs and their imposters, Master’s thesis, University of Waterloo, (2009)
Peeters, R.: On the \(p\)-ranks of the adjacency matrices of distance-regular graphs. J. Algebraic Combin. 15, 127–149 (2002)
Peisert, W.: All self-complementary symmetric graphs. J. Algebra 240, 209–229 (2001)
Rodrigues, B.G.: Linear codes with complementary duals related to the complement of the Higman-Sims graph. Bull. Iranian Math. Soc. 43(7), 2183–2204 (2017)
van Dam, E.R., Sotirov, R.: New bounds for the max-k-cut and chromatic number of a graph. Linear Algebra Appl. 488, 216–234 (2016)
van Lint, J.H., Wilson, R.M.: A Course in Combinatorics. Cambridge University Press, Cambridge (1992)
Acknowledgements
B. G. Rodrigues acknowledges research support of the National Research Foundation of South Africa (Grant Numbers 95725 and 106071).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is based on the research supported by the National Research Foundation of South Africa (Grant Numbers 95725 and 106071).
Rights and permissions
About this article
Cite this article
Fish, W., Key, J.D., Mwambene, E. et al. Hamming graphs and special LCD codes. J. Appl. Math. Comput. 61, 461–479 (2019). https://doi.org/10.1007/s12190-019-01259-w
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12190-019-01259-w