Coding with Constraints: Minimum Distance Bounds and Systematic Constructions

TL;DR

This paper investigates error-correcting codes with symbol constraints, deriving bounds on their minimum distance, especially for systematic linear codes, and explores Reed-Solomon subcodes for efficient decoding in distributed storage.

Contribution

It provides theoretical bounds on minimum distance for constrained codes and conditions for achieving these bounds using Reed-Solomon subcodes, advancing code design for distributed storage.

Findings

Derived bounds on minimum distance based on symbol constraints

Refined bounds for systematic linear codes

Conditions for Reed-Solomon subcodes achieving bounds

Abstract

We examine an error-correcting coding framework in which each coded symbol is constrained to be a function of a fixed subset of the message symbols. With an eye toward distributed storage applications, we seek to design systematic codes with good minimum distance that can be decoded efficiently. On this note, we provide theoretical bounds on the minimum distance of such a code based on the coded symbol constraints. We refine these bounds in the case where we demand a systematic linear code. Finally, we provide conditions under which each of these bounds can be achieved by choosing our code to be a subcode of a Reed-Solomon code, allowing for efficient decoding. This problem has been considered in multisource multicast network error correction. The problem setup is also reminiscent of locally repairable codes.