Chordal Construction of Column-Twisted Reed--Solomon Codes
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) Ck−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 Ck−1 of PG(k−1,q). By a chord of Ck−1, we mean a line which is either a tangent, a real secant, or an imaginary secant of Ck−1. Roth and Lempel constructed non-GRS MDS [n,k]q codes by considering projective systems consisting of (n−1) points on Ck−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 Ck−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 ≤ n ≤ (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 Ck−1 together with a pair of Galois conjugate points on an imaginary secant. For odd q, these codes have lengths satisfying k+4 ≤ n≤ (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≤ k≤ (q−3)/2. We use a geometric argument to establish the non-GRS property of CTRS codes for the range 3≤ k ≤ 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≤ k ≤ n/2. We also determine the structure of the dual of any CTRS code.
Related
- Constructions of LCPs and LCD codes from twisted Reed-Solomon codes
- New families of additive non-RS MDS codes and ACD codes from twisted Reed-Solomon codes
- Two variants of Twisted Reed-Solomon Codes
- Generalized Hamming Weights and Subcode Support Weight Distributions of Twisted Reed-Solomon Codes
- Chordal Codes: Code Construction and Efficient Gaussian-Elimination Decoding
BibTeX
@misc{bhagat2026chordal,
title = {{Chordal Construction of Column-Twisted Reed--Solomon Codes}},
author = {Anuj Kumar Bhagat and Krishna Kaipa},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31257},
eprint = {2610.31257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31257},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Chordal Construction of Column-Twisted Reed--Solomon Codes
AU - Anuj Kumar Bhagat
AU - Krishna Kaipa
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31257
UR - https://arxiv.org/abs/2610.31257
AB - 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 ≤ n ≤ (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 ≤ n≤ (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≤ k≤ (q−3)/2. We use a geometric argument to establish the non-GRS property of CTRS codes for the range 3≤ k ≤ 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≤ k ≤ n/2. We also determine the structure of the dual of any CTRS code.
ER -