The combinatorics of LCD codes: linear programming bound and orthogonal matrices

Abstract

Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings R k . We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.

Some new bounds on LCD codes over finite fields

“…In this section we consider binary codes. Recently, Galvez et al [22] obtain the exact values of LD(n, k) for k = 2 and arbitrary n. By Theorem 1 in [22], we know that there exist LCD codes with LD(n, 2) = ⌊ 2n 3 ⌋ only for n ≡ 1, ±2, 3 (mod 6). An [n, k, d] linear code is optimal if the minimum distance achieve the Gresmer Bound.…”

“…The combinatorial functions LD(n, k) and LD(n, d) has been introduced and studied by Dougherty et al [22] and Galvez et al [17]. The definitions of LD(n, k) and LD(n, d) as follows, we will use them frequently in the rest of this paper.…”

Do non-free LCD codes over finite commutative Frobenius rings exist?

“…al. developed a linear programming bound on the largest size of an LCD code of given length and minimum distance [9]. Guneri et.…”

LCD codes from weighing matrices

“…In their work, they presented several constructions of LCD codes and showed that non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. For further recent research on the topic we refer the reader to [2,12,15,28]. One of our main challenges is to construct LCD codes over finite fields that also have good error correcting properties.…”