Linear Codes Over $\mathbb F_q$ Are Equivalent to LCD Codes for $q>3$
Claude Carlet, Sihem Mesnager, Chunming Tang, Yanfeng Qi, Ruud Pellikaan · IEEE Transactions on Information Theory · 2018
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 Fq(q > 3) is equivalent to a Euclidean LCD code and any linear code over Fq2(q > 2) is equivalent to a Hermitian LCD code. Consequently an [n, k, d]-linear Euclidean LCD code over Fqwith q > 3 exists if there is an [n, k, d]-linear code over Fqand an [n, k, d]-linear Hermitian LCD code over Fq2with q > 2 exists if there is an [n, k, d]-linear code over Fq2. Hence, when q > 3 (resp. q > 2) q-ary Euclidean (resp. q2-ary Hermitian) LCD codes possess the same asymptotical bound as q-ary linear codes (resp. q2-ary linear codes). This gives a direct proof that every triple of parameters [n, k, d] which is attainable by linear codes over Fqwith q > 3 (resp. over Fq2with 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 q2-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 Gröbner bases. Finally, we present a new approach of constructing LCD codes by extending linear codes.