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- W2300200446 abstract "One of the important open problems in the theory of central simple algebras is to compute the essential dimension of $operatorname{GL}_n/mu_m$, 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=(operatorname{GL}_{n_1} times dots times operatorname{GL}_{n_r})/C , , ] where $C$ is a central subgroup of $operatorname{GL}_{n_1} times dots times operatorname{GL}_{n_r}$. Equivalently, we are interested in the essential dimension of a generic $r$-tuple $(A_1, dots, A_r)$ of central simple algebras such that $operatorname{deg}(A_i) = n_i$ 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 $operatorname{Code}(C)$ 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 $operatorname{ed}(G)$ we obtain are expressed in terms of coding-theoretic parameters of $operatorname{Code}(C)$, such as its weight distribution. Surprisingly, for many groups of the above form the essential dimension becomes easier to estimate when $r geq 3$; 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 $(operatorname{GL}_{n_1} times dots times operatorname{GL}_{n_r})/C$. This description and its corollaries are used throughout the paper." @default.
- W2300200446 created "2016-06-24" @default.
- W2300200446 creator A5086429352 @default.
- W2300200446 date "2014-06-11" @default.
- W2300200446 modified "2023-09-27" @default.
- W2300200446 title "Essential dimension and linear codes" @default.
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- W2300200446 doi "https://doi.org/10.14288/1.0167327" @default.
- W2300200446 hasPublicationYear "2014" @default.
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