On the Sparseness of Certain MRD Codes

Heide Gluesing-Luerssen

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arXiv · 1906.11691

On the Sparseness of Certain MRD Codes

Heide Gluesing-Luerssen

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Abstract

We determine the proportion of $[3\times 3;3]$-MRD codes over ${\mathbb F}_q$ within the space of all $3$-dimensional $3\times3$-rank-metric codes over the same field. This shows that for these parameters MRD codes are sparse in the sense that the proportion tends to $0$ as $q\rightarrow\infty$. This is so far the only parameter case for which MRD codes are known to be sparse. The computation is accomplished by reducing the space of all such rank-metric codes to a space of specific bases and subsequently making use of a result by Menichetti (1973) on 3-dimensional semifields.

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Heide Gluesing-Luerssen. 2019-06-27. On the Sparseness of Certain MRD Codes. https://arxiv.org/abs/1906.11691

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