Some new ternary linear codes

Year 2017, Volume: 4 Issue: 3, 227 - 234, 15.09.2017

Rumen Daskalov Plamen Hristov

https://doi.org/10.13069/jacodesmath.327360

Abstract

Let an $[n,k,d]_q$ code be a linear code of length $n$, dimension $k$ and minimum Hamming distance $d$ over $GF(q)$. One of the most important problems in coding theory is to construct codes with optimal minimum distances. In this paper 22 new ternary linear codes are presented. Two of them are optimal. All new codes improve the respective lower bounds in [11].

Keywords

Ternary linear codes, Construction X

References

• [1] N. Aydin, I. Siap, D. Ray-Chaudhuri, The structure of 1–generator quasi–twisted codes and new linear codes, Des. Codes Cryptogr. 24(3) (2001) 313–326.
• [2] A. E. Brouwer, Bounds on the Size of Linear Codes, in Handbook of Coding Theory, V.S. PLess, W.C. Huffman, R.A. Brualdi(eds), Elsevier Amsterdam, 1998.
• [3] E. Z. Chen, Database of quasi–twisted codes, available at http://www.tec.hkr.se/~chen/ research/ codes/searchqc2.htm
• [4] E. Z. Chen, A new iterative computer search algorithm for good quasi–twisted codes, Des. Codes Cryptogr. 76(2) (2015) 307–323.
• [5] E. Chen, N. Aydin, A database of linear codes over $F_{13}$ with minimum distance bounds and new quasi–twisted codes from a heuristic search algorithm, J. Algebra Comb. Discrete Appl. 2(1) (2015) 1–16.
• [6] E. Chen, N. Aydin, New quasi–twisted codes over $F_{11}$-minimum distance bounds and a new database, J. Inf. Optim. Sci. 36(1–2) (2015) 129–157.
• [7] R. N. Daskalov, T. A. Gulliver, New good quasi–cyclic ternary and quaternary linear codes, IEEE Trans. Inform. Theory 43(5) (1997) 1647–1650.
• [8] R. Daskalov, P. Hristov, New one–generator quasi–cyclic codes over GF(7), Problemi Peredachi Informatsii 38(1) (2002) 59–63. English translation: Probl. Inf. Transm. 38(1) (2002) 50–54.
• [9] R. Daskalov, P. Hristov, New quasi–twisted degenerate ternary linear codes, IEEE Trans. Inform. Theory 49(9) (2003) 2259–2263.
• [10] R. Daskalov, P. Hristov, E. Metodieva, New minimum distance bounds for linear codes over GF(5), Discrete Math. 275(1–3) (2004) 97–110.
• [11] M. Grassl, Linear code bound [electronic table; online], available at http://www.codetables.de.
• [12] P. P. Greenough, R. Hill, Optimal ternary quasi–cyclic codes, Des. Codes Cryptogr. 2(1) (1992) 81–91.
• [13] T. A. Gulliver, P. R. J. Ostergard, Improved bounds for ternary linear codes of dimension 7, IEEE Trans. Inform. Theory 43(4) (1997) 1377–1381.
• [14] I. Siap, N. Aydin, D. Ray-Chaudhury, New ternary quasi–cyclic codes with better minimum distances, IEEE Trans. Inform. Theory 46(4) (2000) 1554–1558.
• [15] A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory 43(6) (1997) 1757–1766.