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

[摘要] The binary codes C-n (where n is an element of [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 2(F4(2)) are examined. These codes are obtained from the row span of the incidence matrices of the neighbourhood designs of graphs Gamma, Gamma(R), (Gamma) over tilde, (Gamma) over tilde (R), and Gamma(S) defined by the union of the orbits of 2(F4(2)). We prove that dim(C-1756) = 29, dim(C-2304) = 28, C-1756 superset of C-2304 and Ru acts irreducibly on C-2304. Furthermore we have C-1755 = C-2305 = C-4059 = V-4060 (F-2). Aut(D-1755) = Aut(D-1756) = Aut(D-2304) = Aut(D-2305) = Aut(C-1756) = Aut(C-2304) = Ru while Aut(D-4059) = Aut(C-1755) = Aut(C-2305) = Aut(C-4059) = S-4060. We also determine the weight distribution of C-1756 and C-2304 and that of their duals. For each word w(l) of weight l, in the codes C-Gamma 1756 or C-2304 we determine the stabilizer (Ru)w(l), and some primitive designs held by particular codewords. (c) 2012 Elsevier Inc. All rights reserved.

[发布日期] 2012-12-15

[关键词] Design;Code;Sporadic simple group;Rudvalis group;Primitive permutation representation