A self-orthogonal doubly even code invariant under McL:2

We examine a design D and a binary code C constructed from a primitive permutation representation of degree 275 of the sporadic simple group McL. We prove that Aut(C)=Aut(D)=McL:2 and determine the weight distribution of the code and that of its dual. In Section 5, we show that for a word wi of weight i, where i∈{100,112,164,176} the stabilizer (McL)wi is a maximal subgroup of McL. The words of weight 128 splits into three orbits C(128)1, C(128)2 and C(128)3, and similarly the words of weights 132 produces the orbits C(132)1 and C(132)2. For wi∈{C(128)1,C(128)2,C(132)1}, we prove that (McL)wi is a maximal subgroup of McL. Further in Section 6, we deal with the stabilizers (McL:2)wi by extending the results of Section 5 to McL:2.