Lexicodes over Finite Principal Left Ideal Rings

TL;DR

This paper extends the concept of lexicodes to finite principal left ideal rings, analyzing how ordering choices affect code maximality and algorithm exhaustiveness.

Contribution

It introduces conditions on element ordering for meaningful lexicode construction over finite principal left ideal rings and explores their properties.

Findings

Total ordering must respect ideal containment for meaningful results

Algorithm is exhaustive only under specific ordering conditions

Lexicode properties depend heavily on ordering and basis choices

Abstract

Let R be a finite principal left ideal ring. Via a total ordering of the ring elements and an ordered basis a lexicographic ordering of the module R^n is produced. This is used to set up a greedy algorithm that selects vectors for which all linear combination with the previously selected vectors satisfy a pre-specified selection property and updates the to-be-constructed code to the linear hull of the vectors selected so far. The output is called a lexicode. This process was discussed earlier in the literature for fields and chain rings. In this paper we investigate the properties of such lexicodes over finite principal left ideal rings and show that the total ordering of the ring elements has to respect containment of ideals in order for the algorithm to produce meaningful results. Only then it is guaranteed that the algorithm is exhaustive and thus produces codes that are maximal with respect to inclusion. It is further illustrated that the output of the algorithm heavily depends on the total ordering and chosen basis.