On the existence of some completely regular codes in Hamming graphs (2312.08360v1)
Abstract: We solve several first questions in the table of small parameters of completely regular (CR) codes in Hamming graphs $H(n,q)$. The most uplifting result is the existence of a ${13,6,1;1,6,9}$-CR code in $H(n,2)$, $n\ge 13$. We also establish the non-existence of a ${11,4;3,6}$-code and a ${10,3;4,7}$-code in $H(12,2)$ and $H(13,2)$. A partition of the complement of the quaternary Hamming code of length~$5$ into $4$-cliques is found, which can be used to construct completely regular codes with covering radius $1$ by known constructions. Additionally we discuss the parameters ${24,21,10;1,4,12}$ of a putative completely regular code in $H(24,2)$ and show the nonexistence of such a code in $H(8,4)$. Keywords: Hamming graph, equitable partition, completely regular code
- Perfect 2222-colorings of Hamming graphs. J. Comb. Des., 29(6):367–396, June 2021. https://doi.org/10.1002/jcd.21771.
- Completely regular codes: Tables. Online: http://crcodes.xyz.
- P. Kaski and P. R. J. Östergård. Classification Algorithms for Codes and Designs, volume 15 of Algorithms Comput. Math. Springer, Berlin, 2006. https://doi.org/10.1007/3-540-28991-7.
- P. Kaski and O. Pottonen. libexact user’s guide, version 1.0. Technical Report 2008-1, Helsinki Institute for Information Technology HIIT, 2008.
- D. S. Krotov. On weight distributions of perfect colorings and completely regular codes. Des. Codes Cryptography, 61(3):315–329, 2011. https://doi.org/10.1007/s10623-010-9479-4.
- D. S. Krotov. On the OA(1536,13,2,7) and related orthogonal arrays. Discrete Math., 343(2):111659/1–11, 2020. https://doi.org/10.1016/j.disc.2019.111659.
- B. D. McKay and A. Piperno. Practical graph isomorphism, II. J. Symb. Comput., 60:94–112, 2014. https://doi.org/10.1016/j.jsc.2013.09.003.
- S. Niskanen and P. R. J. Östergård. Cliquer user’s guide, version 1.0. Tech. Rep. Y48, Communications Laboratory, Helsinki University of Technology, Espoo, Finland, 2003.
- J. Rifà and J. Pujol. Translation-invariant propelinear codes. IEEE Trans. Inf. Theory, 43(2):590–598, 1997. https://doi.org/10.1109/18.556115.
- Complete enumeration of pure-level and mixed-level orthogonal arrays. J. Comb. Des., 18(2):123–140, 2010. https://doi.org/10.1002/jcd.20236.
- A. Yu. Vasil’eva. Local and interweight spectra of completely regular codes and of perfect colorings. Probl. Inf. Transm., 45(2):151–157, 2009. https://doi.org/10.1134/S0032946009020069, translated from Probl. Peredachi Inf., 45(2):84-90, 2009.