Some new binary codes with improved minimum distances

Abstract

It has been well-known that the class of quasi-cyclic (QC) codes contain many good codes. In this paper, a method to conduct a computer search for binary $2$-generator QC codes is presented, and a large number of good $2$-generator QC codes have been obtained. $5$ new binary QC codes that improve the lower bounds on minimum distance are presented. Furthermore, with new $2$-generator QC codes and Construction X, $2$ new improved binary linear codes are obtained. With the standard construction techniques, another $16$ new binary linear codes that improve the lower bound on the minimum distance have also been obtained.

Keywords

• Binary linear codes,Quasi-cyclic codes,Algorithms

References

1. [1] N. Aydin, I. Siap, D. K. Ray–Chaudhuri, The structure of 1–generator quasi–twisted codes and new linear codes, Des. Codes Cryptogr. 24(3) (2001) 313–326.
2. [2] N. Aydin, I. Siap, New quasi–cyclic codes over $F_5$, Appl. Math. Lett. 15(7) (2002) 833–836.
3. [3] W. Bosma, J. Cannon, C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3–4) (1997) 235–265.
4. [4] C. L. Chen, W. W. Peterson, E. J. Weldon Jr., Some results on quasi–cyclic codes, Inform. and Control 15(5) (1969) 407–423.
5. [5] E. Z. Chen, Six new binary quasi–cyclic codes, IEEE Trans. Inform. Theory 40(5) (1994) 1666–1667.
6. [6] E. Z. Chen, New quasi–cyclic codes from simplex codes, IEEE Trans. Inform. Theory 53(3) (2007) 1193–1196.
7. [7] E. Z. Chen, A new iterative computer search algorithm for good quasi–twisted codes, Des. Codes Cryptogr. 76(2) (2015) 307–323.
8. [8] E. Z. 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.

9. [9] E. Z. Chen, Database of quasi–twisted codes, 2017, available at http://www.tec.hkr.se/~chen/ research/ codes
10. [10] E. Z. Chen, New binary h–generator quasi–cyclic codes by augmentation and new minimum distance bounds, Des. Codes Cryptogr. 80(1) (2016) 1–10.
11. [11] R. N. Daskalov, T. A. Gulliver, New good quasi–cyclic ternary and quaternary linear codes, IEEETrans. Inform. Theory 43(5) (1997) 1647–1650.
12. [12] R. Daskalov, P. Hristov, Some new quasi–twisted ternary linear codes, J. Algebra Comb. Discrete Appl. 2(3) (2015) 211–216.
13. [13] M. Grassl, Bounds on the minimum distances of linear codes, available at http://www.codetables.de, accessed on November 2, 2016. [14] T. A. Gulliver, V. K. Bhargava, Some best rate 1/p and rate (p-1)/p systematic quasi–cyclic codes, IEEE Trans. Inform. Theory 37(3) (1991) 552–555.
14. [15] T. A. Gulliver, V. K. Bhargava, Nine good rate (m-1)/pm quasi–cyclic codes, IEEE Trans. Inform. Theory 38(4) (1992) 1366–1369.
15. [16] T. A. Gulliver, V. K. Bhargava, Twelve good rate (m-r)/pm quasi–cyclic codes, IEEE Trans. Inform. Theory 39(5) (1993) 1750–1751.
16. [17] T. A. Gulliver, V. K. Bhargava, Two new rate 2/p binary quasi–cyclic codes, IEEE Trans. Inform. Theory 40(5) (1994) 1667–1668.
17. [18] I. Siap, N. Aydin, D. K. Ray–Chaudhuri, New ternary quasi–cyclic codes with better minimum distances, IEEE Trans. Inform. Theory 46(4) (2000) 1554–1558.
18. [19] H. van Tilborg, On quasi–cyclic codes with rate 1/m, IEEE Trans. Inform. Theory 24(5) (1978) 628–630.