Codes with a pomset metric and constructions

Abstract

Brualdi’s introduction to the concept of poset metric on codes over $$\mathbb {F}_{q}$$ paved a way for studying various metrics on $$\mathbb {F}_{q}^{n}$$ . As the support of vector x in $$\mathbb {F}_{q}^{n}$$ is a set and hence induces order ideals and metrics on $$\mathbb {F}_{q}^{n}$$ , the poset metric codes could not accommodate Lee metric structure due to the fact that the support of a vector with respect to Lee weight is not a set but rather a multiset. This leads the authors to generalize the poset metric structure on to a pomset (partially ordered multiset) metric structure. This paper introduces pomset metric and initializes the study of codes equipped with pomset metric. The concept of order ideals is enhanced and pomset metric is defined. Construction of pomset codes are obtained and their metric properties like minimum distance and covering radius are determined.