For any $$ \lambda \in GF(q)^{\ast} $$ a $$ \lambda $$ -constacyclic code $$ C^{n,q,\lambda } : = \,\langle g(x) \rangle $$ , of length $$ n $$ is a set of polynomials in the ring $$ GF(q)[x]/x^{n} - \lambda $$ , which is generated by some polynomial divisor $$ g(x) $$ of $$ x^{n} - \lambda $$ . In this paper a general expression is presented for the uniquely determined idempotent generator of such a code. In particular, if $$ g(x): = (x^{n} - \lambda) / P_{t}^{n,q,\lambda } (x) $$ , where $$ P_{t}^{n,q,\lambda } (x) $$ is an irreducible factor polynomial of $$ x^{n} - \lambda $$ , one obtains a so-called minimal or irreducible constacyclic code. The idempotent generator of a minimal code is called a primitive idempotent generating polynomial or, shortly, a primitive idempotent. It is proven that for any triple $$ (n,q,\lambda ) $$ with $$ (n,q) = 1 $$ the set of primitive idempotents gives rise to an orthogonal matrix. This matrix is closely related to a table which shows some resemblance with irreducible character tables of finite groups. The cases $$ \lambda = 1 $$ (cyclic codes) and $$ \lambda = - 1 $$ (negacyclic codes), which show this resemblance most clearly, are studied in more detail. All results in this paper are extensions and generalizations of those in van Zanten (Des Codes Cryptogr 75:315–334, 2015).

A snake-in-the-box code (or snake) of word length n is a simple circuit in an n-dimensional cube Q n , with the additional property that any two non-neighboring words in the circuit differ in at least two positions. To construct such snakes a straightforward, non-recursive method is developed based on special linear codes with minimum distance 4. An extension of this method is used for the construction of covers of Q n consisting of 2 m-1 vertex-disjoint snakes, for 2 m-1 < n ≤ 2 m . These covers turn out to have a symmetry group of order 2 m .

In this correspondence, we discuss a modified version of a method due to Bakos and, independently, to Robinson and Cohn for the construction of Gray sequences. We make use of this construction to prove the existence of so-called balanced cyclic half Gray sequences. Furthermore, we discuss a specific type of counting sequences, called maximum counting sequences, and we prove a conjecture of Robinson and Cohn with respect to the existence of balanced maximum counting sequences.

In this paper, we present some results concerning a decomposition of binary self-dual codes having an automorphism of an order which is the product of two odd prime numbers. These results are applied to construct self-dual [72,36,12] codes with an automorphism of order 15. Furthermore, it is proved that the automorphism group of a putative binary extremal self-dual [72,36,16] code contains at most 28 nontrivial types of automorphisms of odd order. A complete list of these possible types of automorphisms is presented.

A sharp lower bound is derived for the cyclic list distance between two codewords, having Hamming distance m, in the standard N-ary Gray code of length n, for 1/spl les/m/spl les/n and for even values of N. The bound generalizes a similar result in the binary case.

A linear code is said to be in minimal-change order if each codeword differs from its predecessor by a word of minimum weight. A rule is presented for constructing such an order in the case in which the code has a basis of codewords with minimum weight. Some consequences concerning the ranking and separability are mentioned.>

A number system is developed for the conversion of natural numbers to the codewords of the Gray code G(n,k) of length n and weight k, and vice versa. The focus is on the subcode G(n,k) of G(n) consisting of those words of G(n) with precisely k 1-bits, 0>