Construction of self-dual matrix codes. (English) Zbl 1476.94035
Summary: 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.
MSC:
| 94B05 | Linear codes (general theory) |
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