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PreprintCC BY 4.0arXiv

Comments on "Multidimensional Costas Arrays and Their Periodicity"

  • Shiv Munagala
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Abstract

Rubio and Torres conjectured that a multidimensional Costas array, defined by a bijection between two boxes of integer points whose domain has at least as many dimensions as its image, can stay Costas in every window of its periodic extension only if its order is a power of two fixed by the dimension of the domain. We give a four-dimensional periodic Costas array of order $6$, with a two-dimensional domain and a two-dimensional image, so the conjecture is false. The verification reduces to $36$ windows. It has been confirmed by computer and in the Lean proof assistant. The array also meets with equality the pigeonhole bound that Rubio and Torres use for their non-existence results, so that bound is tight.

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Cite

BibTeXRIS
BibTeX
@misc{munagala2026comments,
  title = {{Comments on "Multidimensional Costas Arrays and Their Periodicity"}},
  author = {Shiv Munagala},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.00098},
  eprint = {2611.00098},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.00098},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Comments on "Multidimensional Costas Arrays and Their Periodicity"
AU  - Shiv Munagala
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.00098
UR  - https://arxiv.org/abs/2611.00098
AB  - Rubio and Torres conjectured that a multidimensional Costas array, defined by a bijection between two boxes of integer points whose domain has at least as many dimensions as its image, can stay Costas in every window of its periodic extension only if its order is a power of two fixed by the dimension of the domain. We give a four-dimensional periodic Costas array of order $6$, with a two-dimensional domain and a two-dimensional image, so the conjecture is false. The verification reduces to $36$ windows. It has been confirmed by computer and in the Lean proof assistant. The array also meets with equality the pigeonhole bound that Rubio and Torres use for their non-existence results, so that bound is tight.
ER  -