New lower bounds on error-correcting ternary, quaternary and quinary codes. (English) Zbl 1429.94090
Barbero, Ángela I. (ed.) et al., Coding theory and applications. 5th international castle meeting, ICMCTA 2017, Vihula, Estonia, August 28–31, 2017. Proceedings. Cham: Springer. Lect. Notes Comput. Sci. 10495, 228-237 (2017).
Summary: Let \(A_q(n,d)\) denote the maximum size of a \(q\)-ary code with size \(n\) and minimum distance \(d\). For most values of \(n\) and \(d\), only lower and upper bounds on \(A_q(n,d)\) are known. In this paper we present 19 new lower bounds where \(q \in \{3,4,5\}\). The bounds are based on codes whose automorphisms are prescribed by transitive permutation groups. An exhaustive computer search was carried out to find the new codes.
For the entire collection see [Zbl 1370.94004].
For the entire collection see [Zbl 1370.94004].
References:
| [1] | Agrell, E.; Vardy, A.; Zeger, K., A table of upper bounds for binary codes, IEEE Trans. Inform. Theory, 47, 3004-3006, 2001 · Zbl 1003.94045 · doi:10.1109/18.959279 |
| [2] | Assmus, EF; Mattson, HF, New 5-designs, J. Combin. Theory, 6, 122-151, 1969 · Zbl 0179.02901 · doi:10.1016/S0021-9800(69)80115-8 |
| [3] | Assmus, EF; Mattson, HF, On weights in quadratic-residue codes, Discrete Math., 3, 1-20, 1972 · Zbl 0305.94013 · doi:10.1016/0012-365X(72)90021-0 |
| [4] | Best, M.; Brouwer, AE; MacWilliams, FJ; Odlyzko, AM; Sloane, NJA, Bounds for binary codes of length less than 25, IEEE Trans. Inform. Theory, 24, 81-93, 1978 · Zbl 0369.94011 · doi:10.1109/TIT.1978.1055827 |
| [5] | Bogdanova, GT; Brouwer, AE; Kapralov, SN; Östergård, PRJ, Error-correcting codes over an alphabet of four elements, Des. Codes Cryptogr., 23, 333-342, 2001 · Zbl 1011.94029 · doi:10.1023/A:1011275112159 |
| [6] | Boukliev, I.; Kapralov, S.; Maruta, T.; Fukui, M., Optimal linear codes of dimension 4 over \(F_5\), IEEE Trans. Inform. Theory, 43, 308-313, 1997 · Zbl 0871.94046 · doi:10.1109/18.567723 |
| [7] | Bogdanova, GT; Östergård, PRJ, Bounds on codes over an alphabet of five elements, Discrete Math., 240, 13-19, 2001 · Zbl 1005.94032 · doi:10.1016/S0012-365X(00)00383-6 |
| [8] | Brouwer, AE; Hämäläinen, HO; Östergård, PRJ; Sloane, NJA, Bounds on mixed binary/ternary codes, IEEE Trans. Inform. Theory, 44, 140-161, 1998 · Zbl 0911.94009 · doi:10.1109/18.651001 |
| [9] | Cannon, JJ; Holt, DF, The transitive permutation groups of degree 32, Exp. Math., 17, 307-314, 2008 · Zbl 1175.20004 · doi:10.1080/10586458.2008.10129046 |
| [10] | Código [pseud.], A discussion on Foros de Free1X2.com, cited by Andries Brouwer at https://www.win.tue.nl/ aeb/codes/ternary-1.html |
| [11] | Conway, JH; Pless, V.; Sloane, NJA, Self-dual codes over GF(3) and GF(4) of length not exceeding 16, IEEE Trans. Inform. Theory, 25, 312-322, 1979 · Zbl 0401.94025 · doi:10.1109/TIT.1979.1056047 |
| [12] | Elssel, K.; Zimmermann, K-H, Two new nonlinear binary codes, IEEE Trans. Inform. Theory, 51, 1189-1190, 2005 · Zbl 1296.94174 · doi:10.1109/TIT.2004.842767 |
| [13] | Hulpke, A., Constructing transitive permutation groups, J. Symbolic Comput., 39, 1-30, 2005 · Zbl 1131.20003 · doi:10.1016/j.jsc.2004.08.002 |
| [14] | Kaikkonen, MK, Codes from affine permutation groups, Des. Codes Cryptogr., 15, 183-186, 1998 · Zbl 0917.94016 · doi:10.1023/A:1008315701179 |
| [15] | Kschischang, FR; Subbarayan, P., Some ternary and quaternary codes and associated sphere packings, IEEE Trans. Inform. Theory, 38, 227-246, 1992 · Zbl 0744.94024 · doi:10.1109/18.119683 |
| [16] | Laaksonen, A., Östergård, P.R.J.: Constructing error-correcting binary codes using transitive permutation groups (submitted). Preprint at. arXiv:1604.06022 |
| [17] | MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes, North-Holland, Amsterdam (1977) · Zbl 0369.94008 |
| [18] | Niskanen, S., Östergård, P.R.J.: Cliquer User’s Guide, Version 1.0, Communications Laboratory, Helsinki University of Technology, Espoo, Finland, Technical report T48 (2003) |
| [19] | Östergård, PRJ, Two new four-error-correcting binary codes, Des. Codes Cryptogr., 36, 327-329, 2005 · Zbl 1158.94419 · doi:10.1007/s10623-004-1723-3 |
| [20] | Plotkin, M., Binary codes with specified minimum distance, IRE Trans. Inform. Theory, 6, 445-450, 1960 · doi:10.1109/TIT.1960.1057584 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
