{
  "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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31257",
  "author_total": 2,
  "authors": [
    {
      "name": "Anuj Kumar Bhagat"
    },
    {
      "name": "Krishna Kaipa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ejqt64dmqksxm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31257",
  "fulltext_url": null,
  "id": "ejqt64dmqksxm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31257",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223322161",
  "pdf_url": "https://arxiv.org/pdf/2610.31257",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ejqt64dmqksxm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31257",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31257"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Chordal Construction of Column-Twisted Reed--Solomon Codes",
  "title_text": "Chordal Construction of Column-Twisted Reed--Solomon Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/ejqt64dmqksxm",
  "venue": "arXiv",
  "versions": []
}