Essential dimension and error-correcting codes
Shane Cernele, Zinovy B. Reichstein · Pacific Journal of Mathematics · 2015
One of the important open problems in the theory of central simple algebras is to compute the essential dimension of GL n = 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 n 1 GL n r .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 / D 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 n 1 GL n r /=C .This description and its corollaries are used throughout the paper.