Coding Theory

Coding theory lies naturally at the intersection of a large number of disciplines in pure and applied mathematics: algebra and number theory, probability theory and statistics, communication theory, discrete mathematics and combinatorics, complexity theory, and statistical physics. The workshop on coding theory covered many facets of the recent research advances

δ-Sequences and Evaluation Codes de ned by Plane Valuations at Infinity

We introduce the concept of δ-sequence. A δ-sequence ∆ generates a well-ordered semigroup S in Z2 or R. We show how to construct (and compute parameters) for the dual code of any evaluation code associated with a weight function defined by ∆ from the polynomial ring in two indeterminates to a semigroup S as above. We prove that this is a simple procedure which can be understood by considering a particular class of valuations of function fields of surfaces, called plane valuations at infinity. We also give algorithms to construct an unlimited number of δ-sequences of the diferent existing types, and so this paper provides the tools to know and use a new large set of codes

Skew Constacyclic Codes over Finite Fields and Finite Chain Rings

This paper overviews the study of skew Θ-λ-constacyclic codes over finite fields and finite commutative chain rings. The structure of skew Θ-λ-constacyclic codes and their duals are provided. Among other results, we also consider the Euclidean and Hermitian dual codes of skew Θ-cyclic and skew Θ-negacyclic codes over finite chain rings in general and over Fpm+uFpm in particular. Moreover, general decoding procedure for decoding skew BCH codes with designed distance and an algorithm for decoding skew BCH codes are discussed