ESSENTIAL DIMENSION AND ERROR-CORRECTING CODES

作     者：Cernele, Shane Reichstein, Zinovy Nguyen, Athena 

作者机构：Univ British Columbia Dept Math 1984 Math Rd Vancouver BC V6T 1Z2 Canada 

出 版 物：《PACIFIC JOURNAL OF MATHEMATICS》 (Pac. J. Math.)

年 卷 期：2015年第279卷第1-2期

页      面：155-179页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：University of British Columbia Natural Sciences and Engineering Research Council of Canada 

主　　题：essential dimension central simple algebra Brauer group error-correcting code Hamming distance 

摘      要：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 = (GL(n1) x ... x GL(nr))/C, where C is a central subgroup of GL(n1) x ... x GL(nr) GLnr. 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) x ... x GL(nr))/C. This description and its corollaries are used throughout the paper.