On Binary LCD Cyclic Codes

New Binary and Ternary LCD Codes

“…As regards binary LCD codes obtained from the univariate case, by using Theorems 16 and 18 we will be able to improve some codes in [17] which are also given in [25]; in this particular case, the main advantages of our procedure are that we can avoid to compute cyclotomic sets (cosets in this case) and that we can get LCD codes with even length. Still on the univariate case but over F 3 , we will provide new examples of ternary LCD codes which are optimal or BKLC.…”

“…That is, for every linear code over a field F q with more than 3 elements, one can construct an LCD code with the same parameters from that code. With respect to binary and ternary LCD codes, the best known LCD codes are reversible and are derived from BCH codes [16,17,25,18]. As it is well-known, subfield-subcodes from codes over large fields can give rise to good codes over small fields.…”

“…Binary J-affine variety codes for one variable with odd length provide reversible codes which essentially coincide with those in [25]; however, our codes are derived from generic results (Theorems 16 and 18) which can simplify some computations. Nevertheless, with one variable, we obtain unknown ternary reversible and binary and ternary nonreversible LCD codes as shown in Section 3.…”

Some binary BCH codes with length n = 2 + 1

“…This result is useful in some cases, such as primitive BCH codes, while useless for BCH codes with length n = q m + 1. Cyclic codes with length n = q m + 1 are LCD codes and contain many good linear codes, see [14], [15] and [19]. Ref.…”

“…Ref. [19] discussed binary LCD cyclic codes of length n ≤ 1025. Known results on the dimensions of BCH codes C(q m + 1, q, δ, 1) and C(q m + 1, q, δ + 1, 0) in [14] and [15] are as follows.…”