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

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.