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SIAM SAGA - Seminar on Applied Geometry and Algebra
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Date: Tuesday, May 10, 2022 at 11:00am Eastern

Speaker: Gretchen Matthews, Virginia Tech

Title: Multivariate Goppa codes

Abstract: Goppa codes were introduced in 1971 by V. D. Goppa using a univariate polynomial g(x), called a generator polynomial, over a finite field. Properties of the Goppa code are tied to those of the generator polynomial. Goppa codes are utilized in the McEliece cryptosystem and can be viewed from several different perspectives, each giving a window into their capabilities. In this talk, we introduce multivariate Goppa codes, which are defined via a multivariate polynomial and contain, as a particular case, the well-known classical Goppa codes. Various descriptions are given to provide insight into the codes, their parameters, and their duals. Moreover, the multivariate Goppa codes can be used to obtain entanglement-assisted quantum error-correcting codes and to build families of long LCD, self-dual, or self-orthogonal codes. (joint work with Hiram Lopez)
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