LCD codes are linear codes that intersect with their dual trivially. Quasi-cyclic codes that are LCD are characterized and studied by using their concatenated structure. Some asymptotic results are derived. Hermitian LCD codes are introduced to that end and their cyclic subclass is characterized. Constructions of QCCD codes from codes over larger alphabets are given.

We introduce a family of algebraic curves over F q 2n (for an odd n) and show that they are maximal. When n = 3, our curve coincides with the F q 6-maximal curve that has been found by Giulietti and Korchmáros recently. Their curve (i.e., the case n = 3) is the first example of a maximal curve proven not to be covered by the Hermitian curve.

We study linear complementary pairs (LCP) of codes (C, D), where both codes belong to the same algebraic code family. We especially investigate constacyclic and quasicyclic LCP of codes. We obtain characterizations for LCP of constacyclic codes and LCP of quasi-cyclic codes. Our result for the constacyclic complementary pairs extends the characterization of linear complementary dual (LCD) cyclic codes given by Yang and Massey. We observe that when C and D are complementary and constacyclic, the codes C and D ⊥ are equivalent to each other. Hence, the security parameter min(d(C), d(D ⊥ )) for LCP of codes is simply determined by one of the codes in this case. The same holds for a special class of quasi-cyclic codes, namely 2D cyclic codes, but not in general for all quasi-cyclic codes, since we have examples of LCP of double circulant codes not satisfying this conclusion for the security parameter. We present examples of binary LCP of quasi-cyclic codes and obtain several codes with better parameters than known binary LCD codes. Finally, a linear programming bound is obtained for binary LCP of codes and a table of values from this bound is presented in the case d(C) = d(D ⊥ ). This extends the linear programming bound for LCD codes.

The Giulietti-Korchmáros (GK) function field is the first example of a maximal function field which is not a subfield of the Hermitian function field over the same constant field. The generalized GK function field Cn was later introduced by Garcia, Güneri and Stichtenoth and was shown to be maximal too. In the present article we determine the automorphism group of the generalized GK function field. We prove that all the automorphisms of Cn fix the unique rational place at infinity and they are exactly the lifts of automorphisms of the Hermitian subfield Hn ⊂ Cn which fix the infinite place of Hn

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