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.

中文翻译：

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自双矩阵码的构建

有限域上的矩阵代码 $${\mathbb {F}}_q$$ 是线性代码，定义为 $$m \times n$$ 矩阵在 $${\mathbb {F}}_q$ 上的向量空间的子空间$. 在本文中，我们展示了如何使用我们称为构建构造的方法从较小尺寸的自对偶矩阵码中获得自对偶矩阵码。我们表明可以使用这种构建结构来构建每个自对偶矩阵代码。利用这个，我们进行分类，即我们找到了一个完整的代表小尺寸自对偶矩阵码的等价类。特别是，我们对大小为 $$2 \times 4$$、$$2 \times 5$$ 超过 $${\mathbb {F}}_{2}$$、大小为 $$2 \ 的自对偶矩阵代码进行了分类3$$, $$2 \times 4$$ 超过 $${\mathbb {F}}_{4}$$, 大小为 $$2 \times 2$$, $$2 \times 3$$ 超过 $${ \mathbb {F}}_{8}$$，大小为 $$2 \times 2$$，