Exchanged hypercubes [Loh et al., IEEE Transactions on Parallel and Distributed Systems 16 (2005) 866–874] are spanning subgraphs of hypercubes with about one half of their edges but still with many desirable properties of hypercubes. Lower and upper bounds on the domination number of exchanged hypercubes are proved which in particular imply that γ(EH(2, t)) = 2 holds for any t ≥ 2. Using Hammi...

We construct a 1-factorization of the complement Σm of the linear Hamming code of length m = mr = 2 r − 1 in the m-cube Qm, for r ≥ 2, having the following equitable property: its component 1-factors intersect each Cayley parallel 1-factor of Qm at a constant number of edges, (namely 2r edges). In the way to that construction, we find an equitable mr−1-factorization of Σm formed by two factors ...

Self-dual codes over Z2 ×Z4 are subgroups of Z2 ×Zβ4 that are equal to their orthogonal under an inner-product that relates to the binary Hamming scheme. Three types of self-dual codes are defined. For each type, the possible values α, β such that there exist a code C ⊆ Z 2 ×Z 4 are established. Moreover, the construction of a Z2Z4-linear code for each type and possible pair (α, β) is given. Fi...

A conflict-avoiding code of length n and weight k is defined as a set C ⊆ {0, 1}n of binary vectors, called codewords, all of Hamming weight k such that the distance of arbitrary cyclic shifts of two distinct codewords in C is at least 2k − 2. In this paper, we obtain direct constructions for optimal conflict-avoiding codes of length n = 16m and weight 3 for any m by utilizing Skolem type seque...

In this paper we show that finite rings for which the extension theorem of MacWilliams is valid for Hamming weight must necessarily be Frobenius. This result makes use of a strategy of Dinh and López-Permouth.

For a prime p with p ≡ 3 (mod 4) and an odd number m, the Bentness of the p-ary binomial function fa,b(x) = Tr1 (ax pm−1) + Tr1(bx pn−1 4 ) is characterized, where n = 2m, a ∈ Fpn , and b ∈ Fp2 . The necessary and sufficient conditions of fa,b(x) being Bent are established respectively by an exponential sum and two sequences related to a and b. For the special case of p = 3, we further characte...

In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determi...

This paper contains results on the generalized Hamming weights for the Goethals and Preparata codes over Z 4 : We give an upper bound on the rth generalized Hamming weights d r (m; j) for the Goethals code G m (j) of length 2 m over Z 4 , when m is odd. We also determine d 3:5 (m; j) exactly. The upper bound is shown to be tight up to r = 3:5. Furthermore we determine the rth generalized Hammin...

We derive a recursive formula determining the weight distribution of the [n = (qm − 1)/(q − 1), n − m, 3] Hamming code H(m, q), when (m, q−1) = 1. Here q is a prime power. The proof is based on Moisio’s idea of using Pless power moment identity together with exponential sum techniques.

Dougherty, Gaborit, Harada, Munemasa, and Solé have previously given an upper bound on the minimum Lee weight of a Type IV self-dual -code, using a similar bound for the minimum distance of binary doubly even self-dual codes. We improve their bound, finding that the minimum Lee weight of a Type IV self-dual -code of length is at most 4 12 , except when = 4, and = 8 when the bound is 4, and = 16...