Linear Complementary Dual codes (LCD) are binary linear codes that meet their dual trivially. We construct LCD codes using orthogonal matrices, self-dual codes, combinatorial designs and Gray map from codes over the family of rings R k . We give a linear programming bound on the largest size of an LCD code of given length and minimum distance. We make a table of lower bounds for this combinatorial function for modest values of the parameters.

This paper investigates the existence, enumeration, and asymptotic performance of self-dual and LCD double circulant codes over Galois rings of characteristic p 2 and order p 4 with p an odd prime. When p ≡ 3 (mod 4), we give a method to construct a duality preserving bijective Gray map from such a Galois ring to Z 2 p 2. Closed formed enumeration formulas for double circulant codes that are self-dual (resp. LCD) are derived as a function of the length of these codes. Using random coding, we obtain families of asymptotically good self-dual and LCD codes over Z p 2 with respect to the metric induced by the standard Fp-valued Gray maps.

Abstract-In this paper we classify all extremal and s-extremal binary self-dual codes of length 38. There are exactly 2744 extremal [38; 19; 8] self-dual codes, two s-extremal [38; 19; 6] codes, and 1730 s-extremal [38; 19; 8] codes. We obtain our results from the use of a recursive algorithm used in the recent classification of all extremal self-dual codes of length 36, and from a generalization of this recursive algorithm for the shadow. The classification of s-extremal [38; 19; 6] codes permits to achieve the classification of all s-extremal codes with d =6 .

We first define a new Gray map from R = Z 4 + uZ 4 to Z 2 4 , where u 2 = 1 and study (1 + 2u)-constacyclic codes over R. Also of interest are some properties of (1 + 2u)-constacyclic codes over R. Considering their Z 4 images, we prove that the Gray images of (1+2u)-constacyclic codes of length n over R are cyclic codes of length 2n over Z 4 . In many cases the latter codes have better parameters than those in the online database of Aydin and Asamov. We also give a corrected version of a table of new cyclic R-codes published byÖzen et al. in Finite Fields and Their Applications, 38, (2016) 27-39.