Resumen
In this article, we introduce a new family of polar codes from evaluation codes, called polar decreasing monomial-Cartesian codes, and prove that families of polar codes with multiple kernels over certain symmetric channels can be viewed as polar decreasing monomial-Cartesian codes. This offers a unified treatment for such codes over any finite field. We define decreasing monomial-Cartesian codes as evaluation codes obtained from a set of monomials closed under divisibility over a Cartesian product and determine their parameters (length, dimension, and minimum distance). We show that the dual of a decreasing monomial-Cartesian code is monomially equivalent to a decreasing monomial-Cartesian code. Polar decreasing monomial-Cartesian codes are then obtained by utilizing decreasing monomial-Cartesian codes whose sets of monomials are closed with respect to a partial order. We prove that any sequence of invertible matrices over an arbitrary field satisfying certain conditions polarizes any channel that is symmetric over the field.
Idioma original | Inglés |
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Número de artículo | 9309243 |
Páginas (desde-hasta) | 3664-3674 |
Número de páginas | 11 |
Publicación | IEEE Transactions on Information Theory |
Volumen | 67 |
N.º | 6 |
DOI | |
Estado | Publicada - jun. 2021 |