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Finite Multiset Codes and Affine Lattice Restrictions

Primarily AI-generated textHuman understanding: no partsmath.CO — Combinatorics

Contributed by Daniel Adam Pyle ↗

Version 1 / Oct 20, 2026 / CC BY 4.0

Abstract

We study constant-weight nonnegative integer vectors under half-L1 distance and compare arbitrary codes with full intersections of affine additive-subgroup cosets with the discrete simplex. A heavy-coordinate decomposition gives the exact reduction A_q(n,n-s) = q + B_q(n,s) to codes whose coordinates are bounded by s. It determines the unrestricted optimum throughout 2n > (q+1)s. For every q ≥ 3 and s ≥ 2, at n = qs−1 and D = (q−1)s−1 the unrestricted and full-coset optima are respectively q+1 and q; we classify all q² unrestricted extremizers. We retain a complete classification of both optima for three coordinates and n < 2D. In the next ternary band, 2D ≤ n < 3D, eight words are attainable precisely when 3n ≥ 8D; equality yields an infinite eight-versus-seven full-coset separation. Exact finite certificates also give a six-versus-five gap at (q,n,D) = (4,5,3) and the quinary single-deletion values A_5(9,2) = 143 and A_5(10,2) = 201. The proofs use intersection bounds and explicit integer affine combinations. The unrestricted decomposition extends known methods, and bibliographic novelty remains unresolved, particularly against a closely related 2024 paper not compared in full.

Provenance statement

The mathematical investigation, proof development, computational searches, verification programs, and manuscript drafting were performed by an OpenAI-powered AI assistant called Lyra. Daniel A. Pyle initiated and directed the project and is submitting the work as an independent researcher. Verification included exact computational certificates and separate AI-led checks; it does not constitute human peer review or proof-assistant formalization. Bibliographic novelty remains unresolved. Reproducibility files are available at https://github.com/dapyle11/finite-multiset-codes.

Tools used

OpenAI
OpenAI-powered assistant (Lyra).

References

  1. M. Kovačević and V. Y. F. Tan, ``Codes in the Space of Multisets---Coding for Permutation Channels with Impairments,'' IEEE Transactions on Information Theory 64(7), 5156--5169 (2018). doi:10.1109/TIT.2017.2789292 (https://doi.org/10.1109/TIT.2017.2789292). https://arxiv.org/abs/1612.08837.DOI
  2. A. Kreindel, I. B. Essayag, and A. L. Zabokritskiy (Yohananov), ``Multiset Deletion Codes: Cyclic Constructions, Bounds, and Exact Results,'' arXiv:2601.05636v2 (2026; revised 13 July). https://arxiv.org/abs/2601.05636v2.arXiv
  3. T. Chen, Y. Ma, and X. Zhang, ``Optimal Codes with Small Constant Weight in ℓ1\ell_1-Metric,'' arXiv:2003.00483v2 (2020; revised 9 October). https://arxiv.org/abs/2003.00483v2.arXiv
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  5. T. Chen, Z. He, and G. Ge, ``Constant ℓ1\ell_1-Weight Codes Under ℓ∞\ell_\infty-Metric,'' IEEE Transactions on Information Theory 70(7), 4928--4945 (2024). doi:10.1109/TIT.2024.3385419 (https://doi.org/10.1109/TIT.2024.3385419). Full-text comparison unresolved.DOI
  6. Z. Lángi, Relative Distance of Points of a Convex Body , doctoral dissertation, ELTE TTK, Mathematics Doctoral School (2003), Chapter 2 and Theorem 6. https://www.math.u-szeged.hu/~zlangi/publications/dissertation.pdf.link

Version history

  1. v1Submitted by Daniel Adam PyleInitial depositCurrentOct 18, 2026