One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GL n /µ m , i.e., the essential dimension of a generic division algebra of degree n and exponent dividing m. In this paper we study the essential dimension of groups of the formwhere C is a central subgroup of GL n1 × · · · × GL nr . Equivalently, we are interested in the essential dimension of a generic r-tuple (A 1 , . . . , A r ) of central simple algebras such that deg(A i ) = n i and the Brauer classes of A 1 , . . . , A r satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of C via the error-correcting code Code(C) which we naturally associate to C. We focus on the case where n 1 , . . . , n r are powers of the same prime. The upper and lower bounds on ed(G) we obtain are expressed in terms of coding-theoretic parameters of Code(C), such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when r ≥ 3; in some cases we even compute the exact value. The Appendix by Athena Nguyen contains an explicit description of the Galois cohomology of groups of the form (GL n1 × · · · × GL nr )/C. This description and its corollaries are used throughout the paper.

Abstract. The representation dimension rdimðGÞ of a finite group G is the smallest positive integer m for which there exists an embedding of G in GL m ðCÞ. In this paper we find the largest value of rdimðGÞ, as G ranges over all groups of order p n , for a fixed prime p and a fixed exponent n d 1.

Abstract. The representation dimension rdim(G) of a finite group G is the smallest positive integer m for which there exists an embedding

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