The average radius of a block code is a parameter that occurs naturally in quantization and steganography. We give asymptotic upper and lower bounds on this parameter. In particular we show that for almost all long codes the normalized average radius equals the normalized covering radius. We survey some special graph-theoretic lower bounds.

Høholdt, van Lint, and Pellikaan used order functions to construct codes by means of Linear Algebra and Semigroup Theory only. However, Geometric Goppa codes that can be represented by this method are mainly those based on just one point. In this correspondence, we introduce the concept of near order function with the aim of generalizing this approach in such a way that a wider family of Geometric Goppa codes can be studied on a more elementary setting.

For a linear code C of length n and dimension k, Wolf (1978) noticed that the trellis state complexity s(C) of C is upper-bounded by w(C):=min(k,n-k). We point out some new lower bounds for s(C). In particular, if C is an algebraic-geometric code, then s(C)/spl ges/w(C)-(g-a), where g is the genus of the underlying curve and a is the abundance of the code.

We compute the complete weight hierarchies of all Hermitian codes. The tools used are the arithmetic of Hermitian curves and the order bound on the generalized Hamming weights.

We compute the order (or Feng-Rao (1994)) bound on the minimum distance of one-point algebraic-geometry codes C/sub /spl Omega//(P, /spl rho//sub t/Q), when the Weierstrass semigroup at the point Q is an Arf 91949) semigroup. The results developed to that purpose also provide the dimension of the improved geometric Goppa codes related to these C/sub /spl Omega// (P, /spl rho//sub t/Q).

We determine the second and third generalized Hamming weights of all Hermitian codes.

We study the generalized Hamming weights of geometric Goppa codes, giving a translation of these weights into arithmetical terms, concerning the arithmetic of the curve used to construct the code. We derive some results and bounds that we apply to some known codes.>

A new way to attack the main conjecture on MDS codes for geometric codes is proposed. In particular, the conjecture for codes arising from curves of genus one or two when the cardinal of the ground field is large enough is proven.>