This paper is devoted to the study of lattices generated by finite Abelian groups. Special species of such lattices arise in the exploration of elliptic curves over finite fields. In the case where the generating group is cyclic, they are also known as the Barnes lattices. It is shown that for every finite Abelian group with the exception of the cyclic group of order four these lattices have a basis of minimal vectors. Another result provides an improvement of a recent upper bound by M. Sha for the covering radius in the case of the Barnes lattices. Also discussed are properties of the automorphism groups of these lattices.

A general upper bound on the minimum distance of Goppa codes is given. It is also shown how to choose divisors so that Goppa codes from fiber products of Kummer covers of the projective line have substantially improved lower bounds for the minimum distance compared to the usual Goppa bound.

In this note, we present improvements of recent asymptotic results on constructions of nonlinear codes using algebraic geometry. The main constructions involve an adaptation of recent constructions by Niederreiter and Oumlzbudak; and Stichtenoth and Xing

We propose a simple construction of algebraic-geometric codes which are subcodes of Goppa codes and which coincide with Goppa codes in many cases. The codes we construct have the advantage that for an explicitly given extension of the rational function field, one can easily obtain explicit bases and therefore an exact formula for the dimension. Furthermore, we show that in many cases good upper and lower bounds for the minimum distance can be obtained

We show that Riemann-Roch spaces of divisors from fiber products of Kummer covers of the projective line, which are invariant with respect to the Galois group, decompose as a direct sum of Riemann-Roch spaces of divisors of the projective line. Consequently, one obtains explicit bases and good upper bounds for the minimum distance of the resulting Goppa codes. This correspondence is a generalization of the work of Xing.