Optimal Ferrers Diagram Rank-Metric Codes

Optimal rank-metric codes in Ferrers diagrams are considered. Such codes consist of matrices having zeros at certain fixed positions and can be used to construct good codes in the projective space. Four techniques and constructions of Ferrers diagram rank-metric codes are presented, each providing optimal codes for different diagrams and parameters.Comment: to be presented in Algebra, Codes, and Networks, Bordeaux, June 16 - 20, 201

On the List-Decodability of Random Linear Rank-Metric Codes

The list-decodability of random linear rank-metric codes is shown to match that of random rank-metric codes. Specifically, an $\mathbb{F}_q$-linear rank-metric code over $\mathbb{F}_q^{m \times n}$ of rate $R = (1-\rho)(1-\frac{n}{m}\rho)-\varepsilon$ is shown to be (with high probability) list-decodable up to fractional radius $\rho \in (0,1)$ with lists of size at most $\frac{C_{\rho,q}}{\varepsilon}$, where $C_{\rho,q}$ is a constant depending only on $\rho$ and $q$. This matches the bound for random rank-metric codes (up to constant factors). The proof adapts the approach of Guruswami, H\aa stad, Kopparty (STOC 2010), who established a similar result for the Hamming metric case, to the rank-metric setting

New Explicit Good Linear Sum-Rank-Metric Codes

Sum-rank-metric codes have wide applications in universal error correction and security in multishot network, space-time coding and construction of partial-MDS codes for repair in distributed storage. Fundamental properties of sum-rank-metric codes have been studied and some explicit or probabilistic constructions of good sum-rank-metric codes have been proposed. In this paper we propose three simple constructions of explicit linear sum-rank-metric codes. In finite length regime, numerous good linear sum-rank-metric codes from our construction are given. Most of them have better parameters than previous constructed sum-rank-metric codes. For example a lot of small block size better linear sum-rank-metric codes over ${\bf F}_q$ of the matrix size $2 \times 2$ are constructed for $q=2, 3, 4$. Asymptotically our constructed sum-rank-metric codes are closing to the Gilbert-Varshamov-like bound on sum-rank-metric codes for some parameters. Finally we construct a linear MSRD code over an arbitrary finite field ${\bf F}_q$ with various matrix sizes $n_1>n_2>\cdots>n_t$ satisfying $n_i \geq n_{i+1}^2+\cdots+n_t^2$ , $i=1, 2, \ldots, t-1$, for any given minimum sum-rank distance. There is no restriction on the block lengths $t$ and parameters $N=n_1+\cdots+n_t$ of these linear MSRD codes from the sizes of the fields ${\bf F}_q$.Comment: 32 pages, revised version, merged with arXiv:2206.0233

Valued rank-metric codes

In this paper, we study linear spaces of matrices defined over discretely valued fields and discuss their dimension and minimal rank drops over the associated residue fields. To this end, we take first steps into the theory of rank-metric codes over discrete valuation rings by means of skew algebras derived from Galois extensions of rings. Additionally, we model projectivizations of rank-metric codes via Mustafin varieties, which we then employ to give sufficient conditions for a decrease in the dimension.Comment: 33 page