This correspondence is a short extension to the previous article Bras-Amorós, 2004. In that work, some results were given on one-point codes related to numerical semigroups. One of the crucial concepts in the discussion was the so-called ν-sequence of a semigroup. This sequence has been used in the literature to derive bounds on the minimum distance as well as for defining improvements on the dimension of existing codes. It was proven in that work that the ν-sequence of a semigroup uniquely determines it. Here this result is extended to another object related to a semigroup, the ⊕ operation. This operation has also been important in the literature for defining other classes of improved codes. It is also proven here that, although the infinite set of values in the ν-sequence (resp. the ⊕ values) uniquely determines the associated semigroup, no finite part of it can determine it, because it is shared by infinitely many semigroups. In that reference the proof of the fact that the ν-sequence of a numerical semigroup uniquely determines it is constructive. The result here presented shows that, however, that construction can not be performed as an algorithm with finite input.
|Journal||IEEE Transactions on Information Theory|
|Publication status||Published - 1 Feb 2007|
- Improved one-point codes
- Numerical semigroup
- One-point codes