Partial Unit Memory codes based on Gabidulin codes

(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. Upper bounds for the free rank distance and the slope of (P)UM codes in the sum rank metric are derived. The construction of PUM codes based on Gabidulin codes achieves the upper bound for the free rank distance.

I. Introduction

In [1], Unit Memory (UM) codes were defined as a special class of rate convolutional codes with memory and overall constraint length . In [2], this idea was extended to Partial Unit Memory (PUM) codes where the memory of the convolutional code is also , but the overall constraint length is .