On the construction of 1-dimensional MDS convolutional Goppa codes

TL;DR

This paper investigates the properties of 1-dimensional convolutional Goppa codes, showing that the MDS property is open and providing algorithms to compute free distance, with applications to algebraic characterization of MDS codes.

Contribution

It establishes the lower-semicontinuity of free distance and explicitly computes algebraic conditions for MDS convolutional Goppa codes.

Findings

MDS property is an open condition in the parameter space.

An algorithm for computing free distance of convolutional codes.

Explicit algebraic equations for MDS Goppa codes.

Abstract

We show that the free distance, as a function on a space parameterizing a family of convolutional codes, is a lower-semicontinuous function and that, therefore, the property of being Maximum Distance Separable (MDS) is an open condition. For a class of convolutional codes, an algorithm is offered to compute the free distance. The behaviour of the free distance by enlargements of the alphabet and by increasing the length is also studied. As an application, the algebraic equations characterizing the subfamily of MDS codes is explicitly computed for families of 1-dimensional convolutional Goppa codes (CGC).