{
  "abstract": "In this paper, we investigate extending of $[n,k,n-k+1]$ non-Reed--Solomon(non-RS) MDS codes to $[n+1,k,n-k+2]$ non-RS MDS codes. We develop a practical iterative algorithm for constructing longer MDS codes while preserving their non-RS structure. The proposed algorithm is inspired by the meet-in-the-middle algorithm in cryptanalysis. On average, it requires at most $O(kq^k)$ multiplications, compared with $O\\left(\\binom{n}{k-1} q^{k-1} k^3\\right)$ multiplications for generic exhaustive search. Applying the proposed algorithm to several known families of MDS codes, including elliptic-curve, twisted Reed--Solomon, column-twisted Reed--Solomon, cyclic, and missing-monomial MDS codes over fields of order at most $32$, we obtain new MDS codes of increased lengths that remain inequivalent to Reed--Solomon codes. In particular, we construct $[9,4,6]$ codes over $\\F_9$ and $\\F_{13}$, each with Schur-square dimension $9$. Over $\\F_{17}$, we obtain two inequivalent $[10,4,7]$ codes with Schur-square dimensions $9$ and $10$, respectively. The latter is the dual of an extended $[10,6,5]$ code, demonstrating that the method also applies when $k>n/2$. For these dimension-four codes, the Schur-square dimensions $9$ and $10$ exceed both that of RS code (Schur-square dimension $7$) and those of specific elliptic-curve and column-twisted Reed--Solomon families (Schur-square dimension $8$). Some of these parameters of non-RS MDS codes have not been reported before.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31023",
  "author_total": 2,
  "authors": [
    {
      "name": "Puyin Wang"
    },
    {
      "name": "Jinquan Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d13ytyzvh1601.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31023",
  "fulltext_url": null,
  "id": "d13ytyzvh1601",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31023",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223380991",
  "pdf_url": "https://arxiv.org/pdf/2610.31023",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d13ytyzvh1601.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31023",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31023"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Extension of MDS Codes over Finite Fields",
  "title_text": "Extension of MDS Codes over Finite Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/d13ytyzvh1601",
  "venue": "arXiv",
  "versions": []
}