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Minimal logarithmic signatures for finite groups of Lie type. (English)
Des. Codes Cryptography 55, No. 2-3, 243-260 (2010).
The paper proposes to use minimal logarithmic signatures for finite groups of Lie type. The method presented in the paper is working for the families $\text{PSL}_n(q)$ and $\text{PSp}_{2n}(q)$ and uses Singer subgroups and the Levi decomposition of parabolic subgroups for these groups. To study the minimal logarithmic signature (MLS) conjecture, the authors give some basic results about LS for an arbitrary subset $A$ of a group $G$. Their aim is to obtain general methods to construct MLS for (simple) groups. By using Remark 3.12, they describe parabolic subgroups, sharply transitive sets and standard Levi subgroups for the corresponding (simple) groups. Then, they use the remark as a tool to create MLSs and other interesting LSs for such groups. They also obtain MLSs for $\text{GL}_n(q)$ and $\text{PGL}_n(q)$ using a slightly different method than {\it W. Lempken} and {\it Tran van Trung} [Exp. Math. 14, No. 3, 257‒269 (2005; Zbl 1081.94038)]. The authors construct MLSs for the groups $\text{GL}_{n}(q)$, $\text{PGL}_n(q)$, $\text{SL}_n(q)$ and $\text{Sp}_{2n}(q)$. The blocks of the LSs are obtained from Singer subgroups of the classical (sub)groups. This also produces a spread in the corresponding projective or polar space. Their methods are general and may prove sufficiently strong as tools for constructing MLSs for all finite simple groups.
Reviewer: Ferruh Özbudak (Ankara)