Weighted Reed–Solomon convolutional codes

Abstract

In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed–Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed–Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields.

Keywords:

• MDP convolutional codes
• vandermonde matrices
• weighted Reed–Solomon convolutional codes

MSC(2010):

• 15B33
• 94B10
• 15B05
• 11T71

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 For monomial with minimum degree in x of $det(G(x,Y,\mathrm{B},\mathrm{\Lambda }))$, we mean the monomial as an element in ${\mathbb{F}}_q[Y][x]$, of the form $p(Y)x^b$, where $p(Y)$ is in the ring of coefficients ${\mathbb{F}}_q[Y]$ and b is the smallest exponent of x involved in the expression of $det(G(x,Y,\mathrm{B},\mathrm{\Lambda }))$.

Additional information

Funding

This work was partially supported by the Swiss National Science Foundation through grants no. 188430 and 187711 and by the Spanish I + D + i project PID2019-108668GB-I00 of MCIN/AEI/10.13039/501100011033 and the I + D + i project VIGROB-287 of the Universitat d’Alacant.