Construction of self-dual matrix codes

Abstract

Matrix codes over a finite field $${\mathbb {F}}_q$$ are linear codes defined as subspaces of the vector space of $$m \times n$$ matrices over $${\mathbb {F}}_q$$ . In this paper, we show how to obtain self-dual matrix codes from a self-dual matrix code of smaller size using a method we call the building-up construction. We show that every self-dual matrix code can be constructed using this building-up construction. Using this, we classify, that is, we find a complete set of representatives for the equivalence classes of self-dual matrix codes of small sizes. In particular we have classifications for self-dual matrix codes of sizes $$2 \times 4$$ , $$2 \times 5$$ over $${\mathbb {F}}_{2}$$ , of size $$2 \times 3$$ , $$2 \times 4$$ over $${\mathbb {F}}_{4}$$ , of size $$2 \times 2$$ , $$2 \times 3$$ over $${\mathbb {F}}_{8}$$ , and of size $$2 \times 2$$ , $$2 \times 3$$ over $${\mathbb {F}}_{13}$$ , all of which have been left open from K. Morrison’s classification.