Symmetry codes over GF(3) and new five-designs

AbstractFor every odd prime power q where q ≡ −1(3) we define a (2q + 2, q + 1) code over the field of three elements. It is shown that all the codes in this family are self orthogonal.For q = 5, the (12, 6) code is equivalent to the extended Golay code. For q = 11, it can be shown that the minimum weight of the (24, 12) code is 9. For q = 17, 23, 29 it is shown, in part by computer, that the minimum weights of the (36, 18), (48, 24), and (60, 30) codes are 12, 15, and 18 respectively.There are 5-designs associated with vectors of certain weights in the (12, 6), (24, 12), (36, 18), (48, 24), and (60, 30) codes. There are new 5-designs associated with the last four codes mentioned. The 5-designs related to the (36, 18) and (60, 30) codes are the first 5-designs found with their parameters.For each q we construct a group P of (2q + 2) × (2q + 2) monomial matrices. We show that P leaves the (2q + 2, q + 1) code in the family invariant, and that P{I, −I} is isomorphic to PGL2(q).We can form a Hadamard matrix by considering the rows of this matrix as certain maximal weight vectors contained in this code. This Hadamard matrix is left invariant by the group P described above