On the classification of linear complementary dual codes

Abstract

We give a complete classification of binary linear complementary dual codes of lengths up to 13 and ternary linear complementary dual codes of lengths up to 10.

Classification of optimal quaternary Hermitian LCD codes of dimension $2$

“…We denote by C(a 0 , a 1 , a 2 , a 3 , a 4 , a 5 ) the code whose generator matrix is G(a 0 , a 1 , a 2 , a 3 , a 4 , a 5 ). By the same argument as in [1], we obtain the following lemma.…”

On the minimum weights of binary linear complementary dual codes

“…0, 0, 1) 2 (0, 0, 1, 0) 3 (0, 0, 1, 1) 4 (0, 0, 1, 2) 5 (0, 1, 0, 0) 6 (0, 1, 0, 1) 7 (0, 1, 0, 2) 8 (0, 1, 1, 0) 9 (0, 1, 1, 1) 10 (0, 1, 1, 2) 11 (0, 1, 2, 0) 12 (0, 1, 2, 1) 13 (0, 1, 2, 2) 14 (1, 0, 0, 0) 15 (1, 0, 0, 1) 16 (1, 0, 0, 2) 17 (1, 0, 1, 0) 18 (1, 0, 1, 1) 19 (1, 0, 1, 2) 20 (1, 0, 2, 0) 21 (1, 0, 2, 1) 22 (1, 0, 2, 2) 23 (1, 1, 0, 0) 24 (1, 1, 0, 1) 25 (1, 1, 0, 2) 26 (1, 1, 1, 0) 27 (1, 1, 1, 1) 28 (1, 1, 1, 2) 29 (1, 1, 2, 0) 30 (1, 1, 2, 1) 31 (1, 1, 2, 2) 32 (1, 2, 0, 0) 33 (1, 2, 0, 1) 34 (1, 2, 0, 2) 35 (1, 2, 1, 0) 36 (1, 2, 1, 1) 37 (1, 2, 1, 2) 38 (1, 2, 2, 0) 39 (1, 2, 2, 1) 40 (1,2,2,2) By the method given in Section 2.1, our exhaustive computer search completes the classification of (unrestricted) ternary [n, 4, d] codes for The unique code D 27 is not LCD.…”

“…In [1], the classification of ternary LCD [n, k] codes was done for n ≤ 10. To test equivalence of ternary codes by a program in the language C, the method given in [1] is also used in this note.…”

“…Proof. By [1,Proposition 5], d 3 (4, 1) = 4. From [1, Table 4], it is known that d 3 (n, n − 3) = 3 for n = 5, .…”

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.…”