Some designs and codes from $L_2(q)$
‎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 ‎{$J_2$}‎, J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput.‎, ‎40 (2002) 143-159.‎, ‎Correction to‎: ‎``Codes‎, ‎designs and graphs from the Janko groups‎ ‎{$J_1$} and {$J_2$}'' [J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput‎., 40 (2002) 143-159]‎, J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput.‎, 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 Information security‎, ‎coding theory and related combinatorics‎, ‎NATO Sci‎. ‎Peace Secur‎. ‎Ser‎. ‎D Inf‎. ‎Commun‎. ‎Secur.‎, ‎IOS Amsterdam‎, 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.