The M LS 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 P SU 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), Des. Codes Cryptogr. 77 (1) (2015) [179][180][181][182][183][184][185][186][187][188][189][190][191] and present a new proof in some special cases of this result. As a consequence, the M LS conjecture is still open.
In this paper, we determine all simple block-transitive 3-designs on 65 points with block stabilizer of order greater than 2, admitting the 2-transitive action of the group [Formula: see text]. As a result, we find 23 3-designs on 65 points that have [Formula: see text] as an automorphism group. Also we construct four 3-([Formula: see text]) designs for [Formula: see text] and two 3-([Formula: see text]) designs for [Formula: see text]. All of these designs have [Formula: see text] as the full automorphism group. Some of these designs are flag and anti-flag-transitive.
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