The most important results on C⊥ t (n, q), the dual code of points and t-spaces in PG(n, q) are presented. We focus on the minimum distance and on the small weight codewords. In the third section, a recent result about the classification of the small weight codewords in C⊥ n−1(n, q), q even, is given.

Let R = GR(p, l) be a Galois ring of characteristic p and cardinality pεl, where p and l are prime integers. First, we give a canonical form decomposition for additive cyclic codes over R. This decomposition is used to construct additive cyclic codes and count the number of such codes, respectively. Thenwe give the trace dual code for each additive cyclic code over R from its canonical form dec...

By the Assmus and Mattson theorem, the codewords of each nontrivial weight in an extremal doubly even self-dual code of length 24m form a self-orthogonal 5-design. In this paper, we study the codes constructed from self-orthogonal 5-designs with the same parameters as the above 5-designs. We give some parameters of a self-orthogonal 5-design whose existence is equivalent to that of an extremal ...

Let w 1 = d, w 2 , ... , w s be the weights of the nonzero codewords in a binary linear [n , k, d] code C, and let w1′ , w2′ , ... , ws ′ ′ be the nonzero weights in the dual code C ⊥ . Let t be an integer in the range 0 < t < d such that there are at most d − t weights wi′ with 0 < wi′ ≤ n − t. Assmus and Mattson proved that the words of any weight w i in C form a t-design. We show that if w 2...

A binary code with the same weight distribution as its dual code is called formally self-dual (f.s.d.). We only consider f.s.d. even codes (codes with only even weight codewords). We show that any formally self-dual even binary code C of length n not divisible by 8 is balanced. We show that the weight distribution of a balanced near-extremal f.s.d. even code of length a multiple of 8 is unique....

Let Fq be a finite field of cardinality q, R = Fq[u]/〈u 〉 = Fq+uFq+u 2 Fq+u 3 Fq (u = 0) which is a finite chain ring, and n be a positive integer satisfying gcd(q, n) = 1. For any δ, α ∈ F×q , an explicit representation for all distinct (δ + αu)-constacyclic codes over R of length n is given, and the dual code for each of these codes is determined. For the case of q = 2 and δ = 1, all self-dua...

We study the dual linear code of points and generators on a nonsingular Hermitian variety H(2n + 1, q2). We improve the earlier results for n = 2, we solve the minimum distance problem for general n, we classify the n smallest types of code words and we characterize the small weight code words as being a linear combination of these n types.

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.