Linear codes resulting from finite group actions

Document Type : Research Paper

Author

Department of Mathematics, University of Cadi Ayyad, Box 63 46000 Route Sidi Bouzid, Safi, Morocco

Abstract

In this article, we use group action theory to define some important ternary linear codes. Some of these codes are self-orthogonal having a minimum distance achieving the lower bound in the previous records. Then, we define two new codes sharing the same automorphism group isomorphic to $C_2 \times M_{11}$ where $M_{11}$ is the Sporadic Mathieu group and $C_{2}$ is a cyclic group of two elements. We also study the natural action of the general linear group $GL (k, 2)$ on the vector space $F_2 ^ k$ to characterize Hamming codes $H_k (2)$ and their automorphism group.

Keywords

Main Subjects

References

[1] W. Cary Huffman, Codes and groups, Handbook of coding theory, I, II, North-Holland, Amsterdam, (1998) 1345–
1440.
[2] F. J. MacWilliams, Permutation decoding of systematic codes, Bell System Tech. J., 43 (1964) 485–505.
[3] L. M. G. M. Tolhuizen and W. J. van, Gils A large automorphism group decreases the number of computations in
the construction of an optimal encoder/decoder pair for a linear block code, IEEE Trans. Inf. Theory, 34 (1988)
333–338.
[4] J. D. Key, Permutation decoding for codes from designs, finite geometries and graphs, Information security, cod-
ing theory and related combinatorics, 172–201, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS,
Amsterdam, 2011.
[5] H.-J. Kroll and R. Vincenti, PD-sets for the codes related to some classical varieties, Discrete Math., 301 (2005)
89–105.
[6] G. Chen and R. Li, Ternary self-orthogonal codes of dual distance three and ternary quantum codes of distance
three, Des. Codes, Cryptogr, 69 (2013) 53–63.
[7] F. Liang, Self-orthogonal codes with dual distance three and quantum codes with distance three over F5 , Quantum
Inf. Process., 12 (2013) 3617–3623.
[8] F. De Clerck and M. Delanote, Two-weight codes, partial geometries and Steiner systems, Des. Codes Cryptogr,
21 (2000) 87–98.
[9] Ph. Delsarte, Weights of linear codes and strongly regular normed spaces, Discrete Math., 3 (1972) 47–64.
[10] R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986) 97–122.
[11] K. Ding and C. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE
Trans. Inform. Theory, 61 (2015) 5835–5842.
[12] W. C. Huffman and V. Pless, Fundamentals of error-correcting codes, Cambridge University Press, Cambridge, 2003.
[13] V. Pless, Introduction to the theory of error-correcting codes, Third edition. Wiley-Interscience Series in Discrete
Mathematics and Optimization, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1998.
[14] D. Joyner and A. Ksir, Automorphism groups of some AG codes, IEEE Trans. Inform. Theory, 52 (2006) 3325–3329.
[15] F. J. MacWilliams and N. J. Sloane, The theory of error-correcting codes, I., II., North-Holland Mathematical
Library, 16, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.
[16] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes. Online available at http:///www.
codetables.de.Accessedon2019-10-05.
[17] https://www.gap-system.org