Chordal construction improves lengths of column-twisted Reed Solomon codes
Chordal Construction of Column-Twisted Reed--Solomon Codes
Information Theory
Summary
Certain error-correcting codes called Reed--Solomon codes are important for data transmission and storage but have known limitations in their structure and length. The authors used geometric ideas involving points and lines on specific curves to build new codes that can be longer and still maintain good error correction. These new codes go beyond previously known constructions by adding pairs of special points called Galois conjugates, improving code parameters. They also provided new ways to prove these codes differ from classical ones, expanding the understanding of code properties.
What this means in practice
- •For digital communication engineers: Construct error-correcting codes with longer lengths to improve data integrity over noisy channels using geometric code designs.
- •For storage system designers: Design codes with enhanced parameters for reliable data protection in storage media beyond classical Reed--Solomon limits.
Authors
Anuj Kumar Bhagat, Krishna Kaipa
Abstract
A $[n,k]_q$ Generalized Reed--Solomon (GRS) code can be represented, up to monomial equivalence, by a projective system consisting of $n$ points on a degree $(k-1)$ rational normal curve (RNC) $C_{k-1}$ in PG$(k-1,q)$. We use the term \textit{Column-Twisted Reed--Solomon (CTRS)} codes broadly for $[n,k]_q$ codes whose projective system contains at least $(k+2)$ points on a $C_{k-1}$ of PG$(k-1,q)$. By a chord of $C_{k-1}$, we mean a line which is either a tangent, a real secant, or an imaginary secant of $C_{k-1}$. Roth and Lempel constructed non-GRS MDS $[n,k]_q$ codes by considering projective systems consisting of $(n-1)$ points on $C_{k-1}$ together with a point on a tangent line. More recently, Liu et al. obtained non-GRS $[n, k]_q$ MDS codes by considering projective systems consisting of $(n-2)$ points on $C_{k-1}$ together with a pair of points on a real secant. In the case when $q$ is odd, the length of these codes satisfies $k+4 \leq n \leq (q+3)/2$. In this article, we construct non-GRS $[n,k]_q$ MDS codes by considering projective systems consisting of $(n-2)$ points on $C_{k-1}$ together with a pair of Galois conjugate points on an imaginary secant. For odd $q$, these codes have lengths satisfying $k+4 \le n\le (q+5)/2$, thereby improving both the attainable lengths and dimension ranges of previous TRS and CTRS constructions. In particular, our code attains the length $(q+5)/2$ for every $3\le k\le (q-3)/2$. We use a geometric argument to establish the non-GRS property of CTRS codes for the range $3\le k \le n-4$, while the traditional method of comparison with the Schur square of GRS codes proves the non-GRS property only for the narrower range $3\le k \le n/2$. We also determine the structure of the dual of any CTRS code.