The Existence of Minimal Logarithmic Signatures for some Finite Simple Unitary Groups

TL;DR

This paper investigates the existence of minimal logarithmic signatures for certain finite simple unitary groups, providing a new proof in specific cases and highlighting that the broader MLS conjecture remains unresolved.

Contribution

It proves the existence of minimal logarithmic signatures for some simple unitary groups and identifies gaps in previous proofs, advancing understanding in this area.

Findings

Established minimal logarithmic signatures for some $PSU_{n}(q)$ groups

Identified gaps in prior proofs of related results

Concluded the MLS conjecture remains open

Abstract

The $MLS$ conjecture states that every finite simple group has a minimal logarithmic signature. The aim of this paper is proving the existence of a minimal logarithmic signature for some simple unitary groups $PSU_{n}(q)$. We report a gap in the proof of the main result of [H. Hong, L. Wang, Y. Yang, Minimal logarithmic signatures for the unitary group $U_n(q)$, \textit{Des. Codes Cryptogr.} \textbf{77} (1) (2015) 179--191] and present a new proof in some special cases of this result. As a consequence, the $MLS$ conjecture is still open.