Linear Codes Over $\mathbb F_q$ Are Equivalent to LCD Codes for $q>3$

Abstract

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual are trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. In this paper, we introduce a general construction of LCD codes from any linear codes. Further, we show that any linear code over Fq(q > 3) is equivalent to an Euclidean LCD code and any linear code over F q 2 (q > 2) is equivalent to a Hermitian LCD code. Consequently an [n, k, d]-linear Euclidean LCD code over Fq with q > 3 exists if there is an [n, k, d]-linear code over Fq and an [n, k, d]-linear Hermitian LCD code over F q 2 with q > 2 exists if there is an [n, k, d]-linear code over F q 2 . Hence, when q > 3 (resp.q > 2) q-ary Euclidean (resp. q 2 -ary Hermitian) LCD codes possess the same asymptotical bound as q-ary linear codes (resp. q 2 -ary linear codes). Finally, we present an approach of constructing LCD codes by extending linear codes.

On the minimum weights of binary linear complementary dual codes

“…(4) There are two inequivalent ternary [54, 4,36] codes. None of the two codes D 54,1 and D 54,2 is LCD.…”

“…Then Proof. The largest minimum weight among (unrestricted) ternary [14,4] code is 8 (see [3]). By Lemma 5 and Table 1…”

“…Recently, the largest minimum weights d 3 (n, 4) among all ternary LCD [n, 4] codes have been determined in [6] for n ≡ 4, 5, 7, 8 (mod 40). In this section, we determine d 3 (n, 4) for n ≡ 11, 14, 16, 17, 20, 24, 29, 30, 33, 36, 39 (mod 40).…”

Binary LCD Codes and Self-Orthogonal Codes via Simplicial Complexes

“…Example 4.5: Let |B| = 4 and |A| = 5 ≤ m. Then C D in Theorem 3.1 is a [16,5,8] self-orthogonal code and C ⊥ D is a [16,11,4] code. According to [11], we conclude that both C D and C ⊥ D are distance optimal.…”

“…For implementations against side-channel and fault injection attacks, a new application of binary LCD codes was found by Carlet and Guilley ( [1], [4]). Since then, LCD codes have attracted wide attention from the coding research community ( [5], [8], [15]- [17], [19], [20]). Carlet et al [8] proved that for q > 3, any q-ary linear code is equivalent to an LCD code over F q ; therefore, it is sufficient to investigate binary LCD codes and ternary LCD codes.…”