Maximum Distance Separable Codes for $b$-Symbol Read Channels

TL;DR

This paper establishes a Singleton-type bound for $b$-symbol read channels, constructs new MDS codes meeting this bound using algebraic methods, and characterizes their existence over finite fields.

Contribution

It introduces a Singleton-type bound for $b$-symbol codes, constructs new MDS codes using projective geometry and constacyclic codes, and determines their existence for specific parameters.

Findings

Established a Singleton-type bound for $b$-symbol codes

Constructed new linear MDS $b$-symbol codes over finite fields

Characterized the existence of such codes for certain parameters

Abstract

Recently, Yaakobi et al. introduced codes for $b$-symbol read channels, where the read operation is performed as a consecutive sequence of $b>2$ symbols. In this paper, we establish a Singleton-type bound on $b$-symbol codes. Codes meeting the Singleton-type bound are called maximum distance separable (MDS) codes, and they are optimal in the sense they attain the maximal minimum $b$-distance. Based on projective geometry and constacyclic codes, we construct new families of linear MDS $b$-symbol codes over finite fields. And in some sense, we completely determine the existence of linear MDS $b$-symbol codes over finite fields for certain parameters.