Weight hierarchies of linear codes of dimension 3 (Q5934170)

As the authors point out in the introduction, the knowledge of the weight hierarchy of linear codes is interesting for the implication it has in several aspects of the practical use of such codes in different scenarios. The problem is known to be difficult and many authors, as those mentioned in the references of the article, have contributed to solving some particular instances of it (codes of certain families like the product codes, Hermitian codes, etc., or codes that satisfy certain conditions like the chain condition, or codes up to a certain dimension). NEWLINENEWLINENEWLINEIn the present paper the authors study the hierarchy of the linear codes of dimension 3 over an arbitrary finite field and, in particular, obtain all the possible weight hierarchies for such codes when the field is of size at most 5. Instead of working directly with the hierarchy, they work with the equivalent concept of Difference Sequence (DS) and divide the problem into two subproblems, namely, finding the DS of codes of dimension 3 that satisfy the chain condition and finding the DS of those which do not satisfy the chain condition. In the first subproblem most of the results had already been obtained by the same authors in a previous unpublished work. Hence, the main novelty of the present paper is the collection of results concerning the second subproblem. The technique the authors use to achieve the results is based on the geometric study of the disposition of points on lines in the projective plane \(\text{PG}(2,q)\).