Automorphism Groups of Higher-Weight Dowling Geometries

The weight k Dowling geometry M(Ek, n) is the restriction of the projective geometry PG(n − 1, q) to the projective points having k or fewer nonzero coordinate entries. These matroids arise when examining the Fundamental Problem of Linear Coding Theory in the context of the Critical Problem of Matroid Theory. In this paper, we study the group of automorphisms of M(Ek, n) for all weights k ≠ 2. For fixed q ≠ 2, the automorphism groups of the Dowling geometries M(E3, n), M(E4, n), ..., M(En − 1, n) are isomorphic. If q = 2 and k ≤ n − 2, then the automorphism group is either the symmetric group on n elements or the symmetric group on n + 1 elements according to whether k is odd or even. We conclude with remarks on the automorphism groups of weight 2 Dowling geometries M(E2, n).