LCD codes over ${\mathbb F}_q $ are as good as linear codes for q at least four

The hull H(C) of a linear code C is defined by H(C) = C ∩ C⊥. A linear code with a complementary dual (LCD) is a linear code with H(C) = {0}. The dimension of the hull of a code is an invariant under permutation equivalence. For binary and ternary codes the dimension of the hull is also invariant under monomial equivalence and we show that this invariant is determined by the extended weight enumerator of the code. The hull of a code is not invariant under monomial equivalence if q ≥ 4. We show that every Fq-linear code is monomial equivalent with an LCD code in case q ≥ 4. The proof uses techniques from Gröbner basis theory. We conclude that if there exists an Fq-linear code with parameters [n, k, d]q and q ≥ 4, then there exists also a LCD code with the same parameters. Hence this holds for optimal and MDS codes. In particular there exist LCD codes that are above the Gilbert-Varshamov bound if q is a square and q ≥ 49 by the existence of such codes that are algebraic geometric. Similar results are obtained with respect to Hermitian LCD codes. ∗Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. E-mail: [email protected] 1 ar X iv :1 70 7. 08 85 6v 1 [ cs .I T ] 2 7 Ju l 2 01 7