Systematic maximum sum rank codes

Abstract

In the last decade there has been a great interest in extending results for codes equipped with the Hamming metric to analogous results for codes endowed with the rank metric. This work follows this thread of research and studies the characterization of systematic generator matrices (encoders) of codes with maximum rank distance. In the context of Hamming distance these codes are the so-called Maximum Distance Separable (MDS) codes and systematic encoders have been fully investigated. In this paper we investigate the algebraic properties and representation of encoders in systematic form of Maximum Rank Distance (MRD) codes and Maximum Sum Rank Distance (MSRD) codes. We address both block codes and convolutional codes separately and present necessary and sufficient conditions for an encoder in systematic form to generate a code with maximum (sum) rank distance. These characterizations are given in terms of certain matrices that must be superregular in a extension field and that preserve superregularity after some transformations performed over the base field. We conclude the work presenting some examples of Maximum Sum Rank convolutional codes over small fields. For the given parameters the examples obtained are over smaller fields than the examples obtained by other authors.

A new rank metric for convolutional codes

“…Although we need an enormous field size in the Theorem 12, these type of constructions can be used to obtain superregular matrices in fields much smaller, but in that case these matrices have to be checked individually for superregularity. This approach was already explored in [19] and [1] and is one avenue of research we are interested to investigate. Another important issue that remains open is to provide not only sufficient (as given in Theorem 8) but also necessary conditions for a given convolutional code to be MRP in terms of superregular matrices.…”

“…The following example illustrates the proprieties mentioned above. [1] α [2] α [3] α [4] α [5] α [6] α [7] α [1] α [2] α [3] α [4] α [5] α [6] α [7] α [8] α [2] α [3] α [4] α [5] α [6] α [7] α [8] α [9] 0 0 0 0 α [0] α [1] α [2] α [3] 0 0 0 0 α [1] α [2] α [3] α [4] 0 0 0 0 α [2] α [3] α [4] α [5] …”

Bounds and Genericity of Sum-Rank-Metric Codes

“…Proof: The equivalence (i)⇔(ii) follows directly from [14], [22], [26]. For (ii)⇔(iii), since U ℓ,k ⊂ A ℓ,k it is sufficient to show, that (iii)⇒(ii).…”

“…In the following lemma, the equivalence (i)⇔(ii) was already studied in a similar form in [14], [22], [26]. and generator matrix G ∈ F k×n q m .…”

On the construction of MRD convolutional codes

“…However, there exist very few algebraic constructions of multi-shot network codes in comparison to the literature on one-shot network rank metric codes. To the best of our knowledge, only one class of maximum rank distance convolutional codes in this setting has been presented in [3,15] based on the construction derived in [1].…”