Some new quasi-twisted ternary linear codes

Abstract

Let $[n,k,d]_q$ code be a linear code of length $n$, dimension $k$ and minimum Hamming distance $d$ over $GF(q)$. One of the basic and  most important problems in coding theory is to construct codes with best possible minimum distances. In this paper seven  quasi-twisted  ternary linear codes are constructed. These codes are new and improve the best known lower bounds on the minimum distance in [6].

Keywords

• Ternary linear codes, Quasi-twisted codes

References

1. R. Ackerman and N. Aydin, New quinary linear codes from quasi-twisted codes and their duals, Appl. Math. Lett., 24(4), 512–515, 2011.
2. S. Ball, Three-dimensional linear codes, Online table, http://www-ma4.upc.edu/∼simeon/. E. Z. Chen, Database of quasi-twisted codes, available at http://moodle.tec.hkr.se/ chen/research/codes/searchqt.htm E. Z. Chen, A new iterative computer search algorithm for good quasi-twisted codes, Des. Codes Cryptogr, 76(2), 307-323, 2014.
3. R. Daskalov and P. Hristov, New quasi-twisted degenerate ternary linear codes, IEEE Trans. Inform. Theory, 49(9), 2259–2263, 2003.
4. M. Grassl, Linear code bound, [electronic table; online], http://www.codetables.de. P. P. Greenough and R. Hill, Optimal ternary quasi-cyclic codes, Des. Codes Cryptogr., 2(1), 81–91, 19 T. A. Gulliver and P. R. J. Ostergard, Improved bounds for ternary linear codes of dimension 7, IEEE Trans. Inform. Theory, 43, 1377–1388, 1997.
5. R. Hill, A first course in coding theory, Oxford Applied Mathematics and Computing Sciences Series, 19 T. Maruta, Griesmer bound for linear codes over finite fields, Online table, http://www.mi.s.osakafu- u.ac.jp/~maruta/griesmer.htm. T. Maruta, M. Shinohara and M. Takenaka, Constructing linear codes from some orbits of projectiv- ities, Discrete Math., 308(5-6), 832–841, 2008.
6. E. Metodieva and N. Daskalova, Generating generalized necklaces and new quasi-cyclic codes, Prob- lemi Peredachi Informatsii, (submitted). I. Siap, N. Aydin and D. Ray-Chaudhury, New ternary quasi-cyclic codes with better minimum distances, IEEE Trans. Inform. Theory, 46(4), 1554–1558, 2000.
7. I. Siap, N. Aydin and D. Ray-Chaudhury, The structure of 1-generator quasi-twisted codes and new linear codes, Des. Codes Cryptogr., 24, 313–326, 2001.
8. S. Dougherty, J. Kim and P. Solé, Open problems in coding theory, Contemporary Mathematics, 634, http://dx.doi.org/10.1090/conm/634/12692, 2015.

9. A. Vardy, The intractability of computing the minimum distance of a code, IEEE Trans. Inform. Theory, 43, 1757–1766, 1997.