{
  "abstract": "Let $\\mathcal M$ be a finite spanning multiset of points in ${\\rm PG}(k-1,q)$, and let ${\\rm Char}_{\\mathcal M}$ be its character function. For $1\\le r\\le k$, we study the maximum number of distinct values attained by ${\\rm Char}_{\\mathcal M}$ on the codimension-$r$ flats under a total-multiplicity budget. The spectrum is subject to two natural upper bounds, whose extremal cases lead respectively to $r$-maximum character spectrum ($r$-MCS) and $r$-full character spectrum ($r$-FCS) systems. We establish the existence of $r$-FCS systems in a broad parameter range, and solve the size-constrained extremal problem completely when $r=k-1$, including the minimum-cost multiplicity profiles. For $1\\le r<k$, put $M_r=\\qbinom{k}{r}$ and $B_r=\\qbinom{k-1}{r}$. The minimum size $ν_r(k,q)$ of an $r$-MCS system satisfies \\[\\left\\lceil\\frac{\\binom{M_r}{2}}{B_r}\\right\\rceil\\le ν_r(k,q)\\le \\binom{M_r-1}{2}+\\frac{q^k-q^{k-r}}{q-1}.\\] The upper bound is at most $\\binom{M_r}{2}$, strictly less when $r\\le k-2$, and exact when $r=k-1$. Its proof applies positive additive labelings to the complement of one flat, whose trace is empty while all remaining traces are distinct. For $r=1$, we improve the known upper bounds on the minimum length of such systems. We also develop global moment identities and local flat-incidence bounds that impose strong structural constraints on large character spectra. Under the projective-system correspondence, complementary character spectra are the support-weight spectra of $r$-dimensional subcodes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.42006",
  "author_total": 3,
  "authors": [
    {
      "name": "Shitao Li"
    },
    {
      "name": "Yang Li"
    },
    {
      "name": "Gaojun Luo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7j1n2ywvhxbfk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.42006",
  "fulltext_url": null,
  "id": "7j1n2ywvhxbfk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.42006",
  "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.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.42006",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7j1n2ywvhxbfk.ris",
  "sources": [
    {
      "first_seen": "2026-11-02T12:21:20Z",
      "native_id": "2610.42006",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.42006"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Extremal character spectra of finite projective systems",
  "title_text": "Extremal character spectra of finite projective systems",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/7j1n2ywvhxbfk",
  "venue": "arXiv",
  "versions": []
}