New Explicit Good Linear Sum-Rank-Metric Codes

Sum-rank-metric codes have wide applications in universal error correction, multishot network coding, space-time coding and the 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 give three simple constructions of explicit linear sum-rank-metric codes. In finite length regime, numerous larger linear sum-rank-metric codes with the same minimum sum-rank distances as the previous constructed codes can be derived from our constructions. For example several better linear sum-rank-metric codes over
${\mathbf{F}}_{q}$
with small block sizes and the matrix size
$2 \times 2$
are constructed for
$q=2, 3, 4$
by applying our construction to the presently known best linear codes. Asymptotically our constructed sum-rank-metric codes are close 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
${\mathbf{F}}_{q}$
with various square matrix sizes
$n_{1}, n_{2}, \ldots, n_{t}$
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
${\mathbf{F}}_{q}$
.