On (Partial) Unit Memory Codes Based on Gabidulin Codes

TL;DR

This paper introduces a new construction of (Partial) Unit Memory codes based on Gabidulin codes utilizing the sum rank metric, achieving optimal free rank distance and enhancing decoding performance in network coding.

Contribution

It presents a novel (P)UM code construction based on Gabidulin codes with a modified sum rank metric, including bounds and explicit code design achieving optimal free rank distance.

Findings

Derived upper bounds for free rank distance and slope in sum rank metric.

Provided an explicit construction of (P)UM codes achieving the upper bound.

Extended the concepts of rank metric to convolutional code design.

Abstract

(Partial) Unit Memory ((P)UM) codes provide a powerful possibility to construct convolutional codes based on block codes in order to achieve a high decoding performance. In this contribution, a construction based on Gabidulin codes is considered. This construction requires a modified rank metric, the so-called sum rank metric. For the sum rank metric, the free rank distance, the extended row rank distance and its slope are defined analogous to the extended row distance in Hamming metric. Upper bounds for the free rank distance and the slope of (P)UM codes in the sum rank metric are derived and an explicit construction of (P)UM codes based on Gabidulin codes is given, achieving the upper bound for the free rank distance.