On the minimum weights of binary linear complementary dual codes

Abstract

It is a fundamental problem to determine the largest minimum weight d 3 (n, k) among all ternary linear complementary dual [n, k] codes. In this note, we determine d 3 (n, 4) for n ≡ 11, 14, 16, 17, 20, 24, 29, 30, 33, 36, 39 (mod 40). We also determine d 3 (n, k), when (n, k) = (121s+17, 5), (364s+13, 6), (364s+18, 6) and (1093s+14, 7), for a nonnegative integer s. In addition, we determine d 3 (n, k) for k = n − 2, n −

Construction for both self-dual codes and LCD codes

“…Let d LCD (n, k) denote the largest minimum weight among all LCD [n, k] code. Galvez, Kim, Lee, Roe and Won [11], Harada and Saito [19], Araya and Harada [1] determined the exact value of d LCD (n, k) for n ≤ 12, 13 ≤ n ≤ 16 and 17 ≤ n ≤ 24, respectively. Bouyuklieva [7] determined the exact value of d LCD (n, k) for n = 25, 27 and gave d LCD (n, k) for n = 26, 28 ≤ n ≤ 40.…”

“…Bouyuklieva [7] determined the exact value of d LCD (n, k) for n = 25, 27 and gave d LCD (n, k) for n = 26, 28 ≤ n ≤ 40. Galvez, Kim, Lee, Roe and Won [11], Harada and Saito [19], Araya and Harada [1], Araya, Harada and Saito [3] determined the exact value of d LCD (n, k) for k = 2, 3, 4, 5, respectively. Also, Dougherty, Kim, Ozkaya, Sok and Solé [10], Araya and Harada [2], Araya, Harada and Saito [4] determined the exact value of d LCD (n, k) for k = n − 1, k ∈ {n − 2, n − 3, n − 4}, k = n − 5, respectively.…”

Optimal Binary LCD Codes

“…The largest minimum weights d LCD (n, k) among all binary LCD [n, k] codes were determined in [7], [10] and [1] for n ≤ 12, 13 ≤ n ≤ 16, and 17 ≤ n ≤ 24, respectively. In this section, we extend the results to lengths up to 40.…”

“…Binary optimal LCD codes have been classified up to length 16 in [10]. In [1], all values of d LCD (n, k) for n ≤ 24 are obtained and some classification results on LCD codes with such lengths are presented. We extend the tables with values of d LCD (n, k) to length 40.…”

Group LCD and Group Reversible LCD Codes

“…In this paper, the authors construct optimal and quasi-optimal examples of binary and ternary LCD codes. Recently in [1], the authors determine the largest minimum weight of LCD [n, 4] codes for n ≡ 2, 3, 5, 6, 9, 10, 13 (mod 15) and the largest minimum weight of LCD [n, 5] codes for n ≡ 3, 4, 5, 7, 11, 19, 20, 22, 26 (mod 31). In another recent paper by S. Bouyuklieva ([3]), optimal binary LCD [n, k] codes are given for k ≤ 32 and n ≤ 40.…”