Essential dimension and error-correcting codes

Shane Cernele and Zinovy Reichstein

Vol. 279 (2015), No. 1-2, 155–179

DOI: 10.2140/pjm.2015.279.155

Abstract

One of the important open problems in the theory of central simple algebras is to compute the essential dimension of ${GL}_n∕{\mu }_m$, 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 form

$$G=\left({GL}_{n_1}\times \cdots \times {GL}_{n_r}\right)∕C,$$

where $C$ is a central subgroup of ${GL}_{n_1}\times \cdots \times {GL}_{n_r}$. Equivalently, we are interested in the essential dimension of a generic $r$-tuple $\left(A_1,\dots ,A_r\right)$ of central simple algebras such that $deg\left(A_i\right)=n_i$ and the Brauer classes of $A_1,\dots ,A_r$ satisfy a system of homogeneous linear equations in the Brauer group. The equations depend on the choice of $C$ via the error-correcting code $Code\left(C\right)$ which we naturally associate to $C$. We focus on the case where $n_1,\dots ,n_r$ are powers of the same prime. The upper and lower bounds on $ed\left(G\right)$ we obtain are expressed in terms of coding-theoretic parameters of $Code\left(C\right)$, such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when $r\ge 3$; 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 $\left({GL}_{n_1}\times \cdots \times {GL}_{n_r}\right)∕C$. This description and its corollaries are used throughout the paper.

To the memory of Robert Steinberg

Keywords

essential dimension, central simple algebra, Brauer group, error-correcting code, Hamming distance

Mathematical Subject Classification 2010

Primary: 20G15, 16K20, 16K50

Secondary: 94B05

Milestones

Received: 27 November 2014

Revised: 25 March 2015

Accepted: 25 March 2015

Published: 21 December 2015