The Weierstrass semigroup H (P) is well known and has been studied. Recently there has been a renewed interest in these semigroups because of applications in coding theory. Generalizations of the Weierstrass semigroup H (P) to n-tuples P 1 , . . . , P n have been made and studied. We will state and study another possible generalization.

The aim of this work is to exhibit some relations between partitions of natural numbers and Arf semigroups. We also give characterizations of Arf semigroups via the hook-sets of Young tableaux of partitions.

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In this study we examine some combinatorial properties of the Arf semigroup. In previous work, the author and Karakaş, Gümüşbaş defined an Arf partition of a positive integer n. Here, we continue this work and give new results on Arf partitions. In particular, we analyze the relation among an Arf partition, its Young dual diagram, and the corresponding rational Young diagram. Additionally, this study contains some results that present the relations between partitions and Arf semigroup polynomials.

In this short note we give an exact count for the number of self-dual codes over a finite field Fq of odd characteristic containing a given self-orthogonal code. This generalizes an analogous result of MacWilliams, Sloane, and Thompson over the field F2 to arbitrary odd finite fields Fq .