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We construct an algorithm which computes the catenary and tame degree of a numerical monoid. As an example we explicitly calculate the catenary and tame degree of numerical monoids generated by arithmetical sequences in terms of their first element, the number of elements in the sequence and the di¤erence between two consecutive elements of the sequence.

We examine the Delta set of a cancellative and reduced atomic monoid S where every set of lengths of the factorizations of each element in S is bounded. In particular, we show the connection between the elements of (S) and the Betti elements of S. We prove how the minimum and maximum element of (S) can be determined using the Betti elements of S. This leads to a determination of when (S) is a singleton. We then apply these results to the particular case where S is a numerical monoid that requires three generators. Conclusions are drawn in the cases where S has a unique minimal presentation, or has multiplicity three.

Abstract. The weight hierarchy of one-point algebraic geometry codes can be estimated by means of the generalized order bounds, which are described in terms of a certain Weierstrass semigroup. The asymptotical behaviour of such bounds for r ≥ 2 differs from that of the classical Feng-Rao distance (r = 1) by the so-called Feng-Rao numbers. This paper is addressed to compute the Feng-Rao numbers for numerical semigroups of embedding dimension two (with two generators), obtaining a closed simple formula for the general case by using numerical semigroup techniques. These involve the computation of the Apéry set with respect to an integer of the semigroups under consideration. The formula obtained is applied to lower-bounding the generalized Hamming weights, improving the bound given by Kirfel and Pellikaan in terms of the classical Feng-Rao distance. We also compare our bound with a modification of the Griesmer bound, improving this one in many cases.

Abstract. We give some general results concerning the computation of the generalized Feng-Rao numbers of numerical semigroups. In the case of a numerical semigroup generated by an interval, a formula for the r th Feng-Rao number is obtained.