Linear Codes Over 𝔽q Are Equivalent to LCD Codes for q>3

Linear codes with complementary duals (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. Nonbinary 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 $\mathbb F_{q} (q>3)$ is equivalent to a Euclidean LCD code and any linear code over $\mathbb F_{q^{2}} (q>2)$ is equivalent to a Hermitian LCD code. Consequently an $[n,k,d]$ -linear Euclidean LCD code over $\mathbb F_{q}$ with $q>3$ exists if there is an $[n,k,d]$ -linear code over $\mathbb F_{q}$ and an $[n,k,d]$ -linear Hermitian LCD code over $\mathbb F_{q^{2}}$ with $q>2$ exists if there is an $[n,k,d]$ -linear code over $\mathbb F_{q^{2}}$ . 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). This gives a direct proof that every triple of parameters $[n,k,d]$ which is attainable by linear codes over $\mathbb F_{q}$ with $q>3$ (resp. over $\mathbb F_{q^{2}}$ with $q>2$ ) is attainable by Euclidean LCD codes (resp. by Hermitian LCD codes). In particular there exist families of $q$ -ary Euclidean LCD codes ( $q>3$ ) and $q^{2}$ -ary Hermitian LCD codes ( $q>2$ ) exceeding the asymptotical Gilbert–Varshamov bound. Further, we give a second proof of these results using the theory of Grobner bases. Finally, we present a new approach of constructing LCD codes by extending linear codes.