We recall why linear codes with complementary duals (LCD codes) play a role in counter-measures to passive and active side-channel analyses on embedded cryptosystems. The rate and the minimum distance of such LCD codes must be as large as possible. We recall the known primary construction of such codes with cyclic codes, and investigate other constructions, with expanded Reed-Solomon codes and generalized residue codes, for which we study the idempotents. These constructions do not allow to reach all the desired parameters. We study then those secondary constructions which preserve the LCD property, and we characterize conditions under which codes obtained by direct sum, direct product, puncturing, shortening, extending codes, or obtained by the Plotkin sum, can be LCD.

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 F _q (q > 3) is equivalent to a 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 F _q with q > 3 exists if there is an [n, k, d]-linear code over F _q 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). This gives a direct proof that every triple of parameters [n, k, d] which is attainable by linear codes over F _q with q > 3 (resp. over 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 Gröbner bases. Finally, we present a new approach of constructing LCD codes by extending linear codes.

Linear complementary dual (LCD) cyclic codes were referred historically to as reversible cyclic codes, which had applications in data storage. Due to a newly discovered application in cryptography, there has been renewed interest in LCD codes. In particular, it has been shown that binary LCD codes play an important role in implementations against side-channel attacks and fault injection attacks. In this paper, we first present a new characterization of binary LCD codes in terms of their orthogonal or symplectic basis. Using such a characterization, we solve a conjecture proposed by Galvez et al. on the minimum distance of binary LCD codes. Next, we consider the action of the orthogonal group on the set of all LCD codes, determine all possible orbits of this action, derive simple closed formulas of the size of the orbits, and present some asymptotic results on the size of the corresponding orbits. Our results show that almost all binary LCD codes are odd-like codes with odd-like duals, and about half of q-ary LCD codes have orthonormal basis, where q is a power of an odd prime.