Column Distances of Convolutional Codes Over &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;${\mathbb Z}_{p^r}$ &lt;/tex-math&gt; &lt;/inline-formula&gt;

Diego Napp,Raquel Pinto,Marisa Toste

doi:10.1109/tit.2018.2870436

Column Distances of Convolutional Codes Over <inline-formula> <tex-math notation="LaTeX">${\mathbb Z}_{p^r}$ </tex-math> </inline-formula>

Diego Napp, Raquel Pinto + Show 1 more

Open Access

https://doi.org/10.1109/tit.2018.2870436

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Abstract

Maximum distance profile codes over finite nonbinary fields have been introduced and thoroughly studied in the last decade. These codes have the property that their column distances are maximal among all codes of the same rate and degree. In this paper, we aim at studying this fundamental concept in the context of convolutional codes over a finite ring. We extensively use the concept of p-encoder to establish the theoretical framework and derive several bounds on the column distances. In particular, a method for constructing (not necessarily free) maximum distance profile convolutional codes over Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> r is presented.

Highlights

Massey and Mittelholzer [19] showed that the most appropriate codes for phase modulation are the linear codes over the residue class ring ZM and this class includes the convolutional codes over ZM, where M is a positive integer
In [24], a bound on the free distance of convolutional codes over Zpr was derived, generalizing the bound given in [25] for convolutional codes over finite fields
Once we recall the definition of free distance [21] and [24], we introduce, for the first time, the concept of column distance of convolutional codes over Zpr

Summary

INTRODUCTION

Massey and Mittelholzer [19] showed that the most appropriate codes for phase modulation are the linear codes over the residue class ring ZM and this class includes the convolutional codes over ZM , where M is a positive integer. In [24], a bound on the free distance of convolutional codes over Zpr was derived, generalizing the bound given in [25] for convolutional codes over finite fields Codes achieving such a bound were called Maximal Distance Separable (or MDS). The concept of Maximum Distance Profile (MDP) convolutional codes over (non-binary) finite fields have been defined and fully studied by Rosenthal et al in [9], [11] and [27] These codes are characterized by the property that their column distances are optimal. We derive upper-bounds on the column distances and provide explicit novel constructions of (not necessarily free) MDP convolutional codes over Zpr. We note that the ring size required to build this class of convolutional codes is in general large.

P -basis and p-dimension k

Block codes over a finite ring

Convolutional codes over a finite ring

COLUMN DISTANCE OF CONVOLUTIONAL CODES OVER A FINITE RING

CONSTRUCTIONS OF MDP CONVOLUTIONAL CODES OVER Zpr