Two Families of LCD BCH Codes

Historically, LCD cyclic codes were referred to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In this paper, we explore two special families of LCD cyclic codes, which are both BCH codes. The dimensions and the minimum distances of these LCD BCH codes are investigated.

[1]  Tadao Kasami,et al.  Some results on the minimum weight of primitive BCH codes (Corresp.) , 1972, IEEE Trans. Inf. Theory.

[2]  Pascale Charpin,et al.  Open problems on cyclic codes , 2009 .

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

[4]  A. Kerber,et al.  Error-correcting linear codes : classification by isometry and applications , 2006 .

[5]  Somphong Jitman,et al.  Complementary Dual Subfield Linear Codes Over Finite Fields , 2016, ArXiv.

[6]  M. Esmaeili,et al.  On complementary-dual quasi-cyclic codes , 2009, Finite Fields Their Appl..

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

[8]  C. Ding Codes From Difference Sets , 2014 .

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

[10]  Susanne Ebersbach,et al.  Designs And Their Codes , 2016 .

[11]  C. M. Melas,et al.  A cyclic code for double error correction , 1960 .

[12]  Elwyn R. Berlekamp,et al.  Algebraic Coding Theory: Revised Edition , 2015 .

[13]  Dian-Wu Yue,et al.  Minimum cyclotomic coset representatives and their applications to BCH codes and Goppa Codes , 2000, IEEE Trans. Inf. Theory.

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

[15]  R. Schoof,et al.  Weight formulas for ternary Melas codes , 1992 .

[16]  Shankar Lal,et al.  A reversible code over GF(q) , 1986, Kybernetika.

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

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

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

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

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

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

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

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

[25]  Henry B. Mann,et al.  On the Number of Information Symbols in Bose-Chaudhuri Codes , 1962, Inf. Control..

[26]  Elwyn R. Berlekamp The enumeration of information symbols in BCH codes , 1967 .

[27]  René Schoof,et al.  Hecke operators and the weight distributions of certain codes , 1991, Journal of Combinatorial Theory.

[28]  James L. Massey,et al.  The Necessary and Sufficient Condition for a Cyclic Code to Have a Complementary Dual , 1999 .

[29]  Pascale Charpin,et al.  Studying the locator polynomials of minimum weight codewords of BCH codes , 1992, IEEE Trans. Inf. Theory.

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

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

[32]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

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

[34]  Daniel Augot,et al.  Idempotents and the BCH bound , 1994, IEEE Trans. Inf. Theory.