{
  "abstract": "Subspace-design codes form a central unifying framework in coding theory, connecting probabilistic existence results and explicit algebraically structured constructions. The two core families are folded Reed-Solomon codes and univariate multiplicity codes, both possessing the subspace-design property. These codes reach list-decoding capacity with polynomial-size lists, but their alphabet size grows with code length and hurts practical efficiency. Goyal, Guruswami, and Hsieh achieved a major breakthrough: they constructed the first explicit subspace-design codes over constant-size alphabets via the Alon-Edmonds-Luby expander framework, settling an open problem and showing subspace-design properties can hold without large alphabets. Despite recent progress, constructing subspace-design codes with low-complexity efficient encoding remains a critical open challenge. This work gives an explicit construction of r-dimensional subspace-design codes over a constant alphabet whose size grows exponentially with r. By comparison, Goyal et al. (FOCS 2026) built r-dimensional subspace-design codes with alphabet size doubly exponential in r squared. Our construction is computationally efficient: encoding scales quadratically with block length and linearly with dimension r in field operations. We use modified algebraic geometry codes, replacing rational evaluation points with higher-degree places to obtain a valid additive code. To confirm parameter bounds, we use Moore matrices to analyze determinants with many zero entries. We also generalize univariate multiplicity codes, originally built from Reed-Solomon codes, to algebraic geometry codes, and prove these AG-multiplicity codes achieve optimal parameters.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.19691",
  "author_total": 1,
  "authors": [
    {
      "name": "Shu Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/29rczxgad6vmm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.19691",
  "fulltext_url": null,
  "id": "29rczxgad6vmm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.19691",
  "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": "W7222354661",
  "pdf_url": "https://arxiv.org/pdf/2610.19691",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/29rczxgad6vmm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.19691",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.19691"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Efficient construction of subspace design codes over constant alphabet",
  "title_text": "Efficient construction of subspace design codes over constant alphabet",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/29rczxgad6vmm",
  "venue": "arXiv",
  "versions": []
}