Some designs and binary codes preserved by the simple group Ru of Rudvalis

The binary codes Cn (where n∈{1755,1756,2304,2305,4059}), defined by the rank-3 primitive permutation representation of degree 4060 of the simple group Ru of Rudvalis on the cosets of the Ree group 2F4(2) are examined. These codes are obtained from the row span of the incidence matrices of the neighbourhood designs of graphs Γ, ΓR, , , and ΓS defined by the union of the orbits of 2F4(2). We prove that dim(C1756)=29, dim(C2304)=28, C1756⊃C2304 and Ru acts irreducibly on C2304. Furthermore we have C1755=C2305=C4059=V4060(F2), Aut(D1755)=Aut(D1756)=Aut(D2304)=Aut(D2305)=Aut(C1756)=Aut(C2304)=Ru while Aut(D4059)=Aut(C1755)=Aut(C2305)=Aut(C4059)=S4060. We also determine the weight distribution of C1756 and C2304 and that of their duals. For each word wl of weight l, in the codes CΓ1756 or C2304 we determine the stabilizer (Ru)wl, and some primitive designs held by particular codewords.