Transactions on Combinatorics (Mar 2014)
Some designs and codes from L_2(q)
Abstract
For $q \in \{7,8,9,11,13,16\}$, we consider the primitive actions of $L_2(q)$ and use Key-Moori Method 1 as described in [Codes, designs and graphs from the Janko groups {$J_1$} and {$J_2$}, {\em J. Combin. Math. Combin. Comput.}, {\bf 40} (2002) 143--159., Correction to: ``Codes, designs and graphs from the Janko groups {$J_1$} and {$J_2$}'' [J. Combin. Math. Combin. Comput. {\bf 40} (2002) 143--159], {\em J. Combin. Math. Combin. Comput.}, {\bf 64} (2008) 153.] to construct designs from the orbits of the point stabilisers and from any union of these orbits. We also use Key-Moori Method 2 (see {\em Information security, coding theory and related combinatorics}, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS Amsterdam, {\bf 29} (2011) 202--230.) to determine the designs from the maximal subgroups and the conjugacy classes of elements of these groups. The incidence matrices of these designs are then used to generate associated binary codes. The full automorphisms of these designs and codes are also determined.
