A Linear Algebraic Approach to Subfield Subcodes of GRS Codes

TL;DR

This paper introduces a linear algebraic method to analyze subfield subcodes of GRS codes, generalizing conjugacy constraints and enabling the construction of nested subcodes with improved error-correcting capabilities.

Contribution

It presents a novel linear algebraic framework for deriving message constraints for subfield subcodes of GRS codes, extending known conjugacy constraints and facilitating nested code design.

Findings

Derived generalized message constraints for subfield subcodes

Demonstrated the ability to construct nested subcodes with higher design distance

Provided a unified algebraic approach applicable to various GRS code subclasses

Abstract

The problem of finding subfield subcodes of generalized Reed-Solomon (GRS) codes (i.e., alternant codes) is considered. A pure linear algebraic approach is taken in order to derive message constraints that generalize the well known conjugacy constraints for cyclic GRS codes and their Bose-Chaudhuri-Hocquenghem (BCH) subfield subcodes. It is shown that the presented technique can be used for finding nested subfield subcodes with increasing design distance.