On the Hasse diagram of binary linear codes

Abstract

A binary [Formula: see text]-linear code [Formula: see text] is a [Formula: see text]-dimensional subspace of [Formula: see text]. For [Formula: see text], the set [Formula: see text] is a coset of [Formula: see text]. In this work, we study a partial ordering on the set of cosets of a binary linear code [Formula: see text] of length [Formula: see text]: for [Formula: see text], [Formula: see text] provided that [Formula: see text], and we construct its associated Hasse diagram. We give general and particular examples that help to understand the concept and we obtain general properties of this graph.