Some new results on Generalized Algebraic Geometric (GAG) codes are obtained. First, we provide some constructions which significantly improve the general lower bounds on the minimum distance of a GAG code. GAG codes associated to specific maximal curves over finite fields are then investigated. As a result, 2895 improvements on MinT's tablesare obtained. Finally, we construct asymptotically good GAG codes with better parameters with respect to those constructed by Spera in 2005. Maximal curves play a role in this context as well.

In this paper, the best parameters of quaternary additive codes of small length are determined using the geometric description. Only one open question remains for length les 13. Among the results obtained in this work are the nonexistence of [12, 7, 5]-codes and [12, 4.5, 7]-codes as well as the existence of a [13, 7.5, 5]-code.

A concept of locally optimal (LO) linear covering codes is introduced in accordance with the concept of minimal saturating sets in projective spaces over finite fields. An LO code is nonshortening in the sense that one cannot remove any column from a parity-check matrix without increasing the code covering radius. Several q/sup m/-concatenating constructions of LO covering codes are described. Taking a starting LO code as a "seed", such constructions produce an infinite family of LO codes with the same covering radius. The infinite families of LO codes are designed using minimal saturating sets as starting codes. New upper bounds on the length function are given. New extremal and classification problems for linear covering codes are formulated and investigated, in particular, the spectrum of possible lengths of LO codes including the greatest possible length. The complete computer classification of the minimal saturating sets in small geometries and of the corresponding LO codes is obtained.