Designs, Codes and Cryptography

, Volume 24, Issue 3, pp 313–326

The Structure of 1-Generator Quasi-Twisted Codes and New Linear Codes

• Nuh Aydin
• Irfan Siap
• Dijen K. Ray-Chaudhuri
Article

Abstract

One of the most important problems of coding theory is to construct codes with best possible minimum distances. Recently, quasi-cyclic (QC) codes have been proven to contain many such codes. In this paper, we consider quasi-twisted (QT) codes, which are generalizations of QC codes, and their structural properties and obtain new codes which improve minimum distances of best known linear codes over the finite fields GF(3) and GF(5). Moreover, we give a BCH-type bound on minimum distance for QT codes and give a sufficient condition for a QT code to be equivalent to a QC code.

quasi-twisted codes new bounds ternary codes

Preview

References

1. 1.
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, New York (1968).Google Scholar
2. 2.
V. K. Bhargava, G. E. Séguin and J. M. Stein, Some (mk, k) cyclic codes in quasi-cyclic form, IEEE Trans. Inform. Theory, Vol. 24, No. 5 (1978) pp. 630–632.Google Scholar
3. 3.
A. E. Brouwer, Linear code bound (server), Eindhoven University of Technology, The Netherlands, http://www.win.tue.nl/math/dw/personalpages/aeb/voorlincod.html.Google Scholar
4. 4.
A. E. Brouwer, Bound on the size of a linear code, Handbook Of Coding Theory (V. S. Pless and W. Huffman, eds.), Vol. 1, Elsevier, New York (1998).Google Scholar
5. 5.
Z. Chen, Six new binary quasi-cyclic codes, IEEE Trans. Inform. Theory, Vol. 40, No. 5 (1994) pp. 1666–1667.Google Scholar
6. 6.
J. Conan and G. E. Séguin, Structural properties and enumeration of quasi-cyclic codes, Appl. Algebra Engrg. Comm. Comput., Vol. 4, No. 1 (1993) pp. 25–39.Google Scholar
7. 7.
R. N. Dasklov, T. A. Gulliver and E. Metodieva, New good quasi-cyclic ternary and quaternary linear codes, IEEE Trans. Inform. Theory, Vol. 43, No. 5 (1997) pp. 1647–1650.Google Scholar
8. 8.
R. N. Daskalov, T. Aaron Gulliver, and E. Metodieva, New ternary linear codes, IEEE Trans. Inform. Theory, Vol. 45, No 5 (1999) pp. 1687–1688.Google Scholar
9. 9.
P. P. Greenough and R. Hill, Optimal ternary quasi-cyclic codes, Des. Codes Cryptogr., Vol. 2, No. 1 (1992) pp. 81–91.Google Scholar
10. 10.
T. A. Gulliver, New optimal ternary linear codes of dimension 6, Ars. Combin., Vol. 40 (1995) pp. 97–108.Google Scholar
11. 11.
T. A. Gulliver and V. K. Bhargava, Some best rate 1/p quasi-cyclic codes over GF(5), Inform. Theory Applic. II, Springer-Verlag, Berlin, New York (1996) pp. 28–40.Google Scholar
12. 12.
T. A. Gulliver, and V. K. Bhargava, Two new rate 2/p binary quasi-cyclic codes, IEEE Trans. Inform. Theory, Vol. 40, No. 5 (1994) pp. 1667–1668.Google Scholar
13. 13.
T. A. Gulliver and V. K. Bhargava, Nine good rate (m - 1)/pm quasi-cyclic codes, IEEE Trans. Inform. Theory, Vol. 38, No 4 (1992) pp. 1366–1369.Google Scholar
14. 14.
T. A. Gulliver and V. K. Bhargava, Some best rate 1/p and rate (p - 1)/p systematic quasi-cyclic codes over GF(3) and GF(4), IEEE Trans. Inform. Theory, Vol. 38, No 4 (1992) pp. 1369–1374.Google Scholar
15. 15.
T. A. Gulliver and V. K. Bhargava, New good rate (m - 1)/pm ternary and quaternary quasi-cyclic codes, Des. Codes Cryptogr., Vol. 7, No. 3 (1996) pp. 223–233.Google Scholar
16. 16.
T. A. Gulliver and P. R. J. Östergard, New binary linear codes, Ars. Comb., Vol. 56 (2000) pp. 105–112.Google Scholar
17. 17.
T. A. Gulliver and P. R. J. Östergard, Improved bounds for ternary linear codes of dimension 7, IEEE Trans. Inform. Theory, Vol. 43, No. 4 (1997) pp. 1377–1381.Google Scholar
18. 18.
R. Hill, An extension theorem for linear codes, Des. Codes Cryptogr., Vol. 17, No. 1-3 (1999) pp. 151–157.Google Scholar
19. 19.
R. Hill and P. P. Greenough, Optimal quasi-twisted codes, in Proceedings of the Second International Workshop Algebraic and Combinatorial Coding Theory, Voneshta Voda, Bulgaria, June (1992) pp. 92–97.Google Scholar
20. 20.
J. M. Jensen, Cyclic concatenated codes with constacyclic outer codes, IEEE Trans. Inform. Theory, Vol. 38, No. 3 (1992) pp. 950–959.Google Scholar
21. 21.
T. Kasami, A Gilbert-Varshamov bound for quasi-cyclic codes of rate 1/2 IEEE Trans. Inform. Theory, Vol. 20 (1974) pp. 679.Google Scholar
22. 22.
A. Krishna and D. V. Sarwate, Pseudocyclic maximum-distance-separable codes, IEEE Trans. Inform. Theory, Vol. 36, No. 4 (1990) pp. 880–884.Google Scholar
23. 23.
K. Lally and P. Fitzpatrick, Construction and classification of quasi-cyclic codes, WCC 99, Workshop on Coding and Cryptography Paris (France), January (1999) pp. 11–14.Google Scholar
24. 24.
F. J. MacWilliams and N. J. A. Sloane, The Theory Of Error Correcting Codes, North Holland, New York (1977).Google Scholar
25. 25.
S. Roman, Field Theory, Springer-Verlag, New York (1995).Google Scholar
26. 26.
S. Roman, Coding and Information Theory, Springer-Verlag, New York (1992).Google Scholar
27. 27.
G. E. Séguin and G. Drolet, The theory of 1-generator quasi-cyclic codes, Manuscript, Dept. Elec. Comput. Eng., Royal Military College of Canada, Kingston, Ont. (1990).Google Scholar
28. 28.
S. E. Tavares, V. K. Bhargava and S. G. S. Shiva, Some rate p/(p + 1) quasi-cyclic codes, IEEE Trans. Inform. Theory, Vol. 20 (1974) pp. 133–135.Google Scholar
29. 29.
K. Thomas, Polynomial approach to quasi-cyclic codes, Bul. Cal. Math. Soc., Vol. 69 (1977) pp. 51–59.Google Scholar
30. 30.
H. van Tilborg, On quasi-cyclic codes with rate 1/m, IEEE Trans. Inform. Theory, Vol. 24, No. 5 (1978) pp. 628–630.Google Scholar