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Modular Constructions of g-Golomb Rulers

  • Aditya Gupta
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Abstract

A set 𝒢 of integers is a g-Golomb ruler if each positive difference appears at most g times between any 2 elements of the set, and G(g,n) denotes the minimum diameter of such a ruler with n marks. We prove a general lemma for passing from certain modular constructions to ordinary g-Golomb rulers. The key point is that, in a modular g-Golomb ruler, no cyclic gap length can occur more than g times. This gives a larger guaranteed cut than the previous average gap argument. We apply this lemma to cyclic relative difference sets, Singer sets, Ruzsa--Spence rulers, and Paley quadratic residues to provide many competing constructions for g-Golomb Rulers. A computation on the grid 1≤ g≤500, n=g+b, 2≤ b≤500, compares the four resulting construction families.

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Cite

BibTeXRIS
BibTeX
@misc{gupta2026modular,
  title = {{Modular Constructions of g-Golomb Rulers}},
  author = {Aditya Gupta},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2607.07931},
  eprint = {2607.07931},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.07931},
}
RIS
TY  - UNPB
TI  - Modular Constructions of g-Golomb Rulers
AU  - Aditya Gupta
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2607.07931
UR  - https://arxiv.org/abs/2607.07931
AB  - A set 𝒢 of integers is a g-Golomb ruler if each positive difference appears at most g times between any 2 elements of the set, and G(g,n) denotes the minimum diameter of such a ruler with n marks. We prove a general lemma for passing from certain modular constructions to ordinary g-Golomb rulers. The key point is that, in a modular g-Golomb ruler, no cyclic gap length can occur more than g times. This gives a larger guaranteed cut than the previous average gap argument. We apply this lemma to cyclic relative difference sets, Singer sets, Ruzsa--Spence rulers, and Paley quadratic residues to provide many competing constructions for g-Golomb Rulers. A computation on the grid 1≤ g≤500, n=g+b, 2≤ b≤500, compares the four resulting construction families.
ER  -