Linear Codes Over $\mathbb F_q$ 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 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.