Ortiz-Ubarri et al.'s elementary-abelian codomain conjecture for circular Costas maps

About 4 years old · traced to 2022

Let $G_1$ and $G_2$ be groups, and let a circular Costas map be a map $\varphi:G_1\to G_2$ that is injective and whose nonzero difference maps are injective; call it standard when its image omits the identity of $G_2$ . A group is elementary abelian if it is a direct product of copies of a cyclic group of prime order. Ortiz-Ubarri et al.'s conjecture. If $\varphi:G_1\to G_2$ is circular Costas, $G_2$ is elementary abelian. The paper states that this conjecture is proved in the indicated sections, so it is resolved.

References

Primary source

Ivelisse Rubio and Jaziel Torres, “Circular Costas maps: a multidimensional analog of circular Costas sequences”, arXiv:2210.16661 (2022).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A 2022 paper reports a proof that every such map has an elementary-abelian target group, but the proof has not been independently verified here.

Ortiz-Ubarri et al.'s conjecture asserts that the target group of every circular Costas map is elementary abelian. The relevant paper identifies this as Conjecture 17 and also as Ortiz-Ubarri et al.'s Conjecture 4.

2022 proof claim

The paper's Theorem $24$ claims that any circular Costas map $\varphi:G_1\to G_2$ has $G_2\cong\mathbb{Z}_p^m$ for some prime $p$ and natural number $m$ . Its argument uses an associated abelian direct-product difference set and results in Sections $4$ and $6$ ; no counterexample, gap report, withdrawal, or independent verification was found.

Current status (as of September 2026): The conjecture is claimed solved by the 2022 paper, but its proof remains unverified in the retrieved record.

Sources

No solutions have been posted yet.