Designed distances and parameters of new LCD BCH codes over finite fields

Let F q $\mathbb {F}_{q}$ be the finite field of q elements and n = q m − 1 with m a positive integer. In this paper we construct a class of BCH and LCD BCH codes of length n over F q $\mathbb {F}_{q}$ and investigate their dimensions and designed distance. Our results show that the designed distances of BCH and LCD BCH codes in this paper are larger than those in [ 11 , Theorems 7, 10, 18, and 22]. It is viewed as a generalized result of [ 11 ].

[1]  Pascale Charpin.  On a class of primitive BCH-codes , 1990, IEEE Trans. Inf. Theory.

[2]  Jon-Lark Kim,et al.  The combinatorics of LCD codes: linear programming bound and orthogonal matrices , 2015, Int. J. Inf. Coding Theory.

[3]  Yue Dianwu,et al.  On the dimension and minimum distance of BCH codes over GF(q) , 1996 .

[4]  O. Antoine,et al.  Theory of Error-correcting Codes , 2022 .

[5]  Xiang Yang,et al.  The condition for a cyclic code to have a complementary dual , 1994, Discret. Math..

[6]  Cunsheng Ding,et al.  Dimensions of three types of BCH codes over GF(q) , 2016, Discret. Math..

[7]  Haode Yan,et al.  Parameters of LCD BCH codes with two lengths , 2018, Adv. Math. Commun..

[8]  Cunsheng Ding,et al.  Parameters of Several Classes of BCH Codes , 2015, IEEE Transactions on Information Theory.

[9]  Cunsheng Ding,et al.  Two Families of LCD BCH Codes , 2016, IEEE Transactions on Information Theory.

[10]  Pradeep Kiran Sarvepalli,et al.  On Quantum and Classical BCH Codes , 2006, IEEE Transactions on Information Theory.

[11]  Cunsheng Ding,et al.  LCD Cyclic Codes Over Finite Fields , 2017, IEEE Transactions on Information Theory.

[12]  James L. Massey.  Reversible Codes , 1964, Inf. Control..

[13]  Carlos R. P. Hartmann,et al.  On the minimum distance of certain reversible cyclic codes (Corresp.) , 1970, IEEE Trans. Inf. Theory.

[14]  James L. Massey,et al.  Linear codes with complementary duals , 1992, Discret. Math..

[15]  W. Cary Huffman,et al.  Fundamentals of Error-Correcting Codes , 1975 .

[16]  Frédérique E. Oggier,et al.  On LCD codes and lattices , 2016, 2016 IEEE International Symposium on Information Theory (ISIT).

[17]  Nicolas Sendrier,et al.  Linear codes with complementary duals meet the Gilbert-Varshamov bound , 2004, International Symposium onInformation Theory, 2004. ISIT 2004. Proceedings..

[18]  Claude Carlet,et al.  Complementary dual codes for counter-measures to side-channel attacks , 2016, Adv. Math. Commun..

[19]  Cunsheng Ding,et al.  The Bose and Minimum Distance of a Class of BCH Codes , 2015, IEEE Transactions on Information Theory.