Some new binary codes with improved minimum distances
DOI:
https://doi.org/10.13069/jacodesmath.404964Keywords:
Binary linear codes, Quasi-cyclic codes, AlgorithmsAbstract
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.
Received: 9 November 2016 Accepted: 22 November 2017Downloads
References
N. Aydin, I. Siap, and D. K. Ray–Chaudhuri, The structure of 1–generator quasi–twisted codes and new linear codes, Des. Codes Cryptogr. 24(3) (2001) 313–326.
N. Aydin and I. Siap, New quasi–cyclic codes over $\mathbb{F}_5$, Appl. Math. Lett. 15(7) (2002) 833–836.
W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24(3–4) (1997) 235–265.
C. L. Chen, W. W. Peterson, and E. J. Weldon Jr., Some results on quasi–cyclic codes, Inform. and Control 15(5) (1969) 407–423.
E. Z. Chen, Six new binary quasi–cyclic codes, IEEE Trans. Inform. Theory 40(5) (1994) 1666–1667.
E. Z. Chen, New quasi–cyclic codes from simplex codes, IEEE Trans. Inform. Theory 53(3) (2007) 1193–1196.
E. Z. Chen, A new iterative computer search algorithm for good quasi–twisted codes, Des. Codes Cryptogr. 76(2) (2015) 307–323.
E. Z. Chen and N. Aydin, A database of linear codes over $\mathbb{F}_{13}$ with minimum distance bounds and new quasi–twisted codes from a heuristic search algorithm, J. Algebra Comb. Discrete Struct. Appl. 2(1) (2015) 1–16.
E. Z. Chen, Database of quasi–twisted codes (2017), available at http://www.tec.hkr.se/~chen/research/codes.
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.
R. N. Daskalov and T. A. Gulliver, New good quasi–cyclic ternary and quaternary linear codes, IEEE Trans. Inform. Theory 43(5) (1997) 1647–1650.
R. Daskalov and P. Hristov, Some new quasi–twisted ternary linear codes, J. Algebra Comb. Discrete Struct. Appl. 2(3) (2015) 211–216.
M. Grassl, Bounds on the minimum distances of linear codes, available at http://www.codetables.de.
T. A. Gulliver and 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.
T. A. Gulliver and V. K. Bhargava, Nine good rate $(m-1)/p^m$ quasi–cyclic codes, IEEE Trans. Inform. Theory 38(4) (1992) 1366–1369.
T. A. Gulliver and V. K. Bhargava, Twelve good rate $(m-r)/p^m$ quasi–cyclic codes, IEEE Trans. Inform. Theory 39(5) (1993) 1750–1751.
T. A. Gulliver and V. K. Bhargava, Two new rate $2/p$ binary quasi–cyclic codes, IEEE Trans. Inform. Theory 40(5) (1994) 1667–1668.
I. Siap, N. Aydin, and D. K. Ray–Chaudhuri, New ternary quasi–cyclic codes with better minimum distances, IEEE Trans. Inform. Theory 46(4) (2000) 1554–1558.
H. C. A. van Tilborg, On quasi–cyclic codes with rate $1/m$, IEEE Trans. Inform. Theory 24(5) (1978) 628–630.
