Polar decreasing monomial-Cartesian codes
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , , , |
|---|---|
| Format: | Preprint |
| Publié: |
2020
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866911771517779968 |
|---|---|
| author | Camps, Eduardo López, Hiram H. Matthews, Gretchen L. Sarmiento, Eliseo |
| author_facet | Camps, Eduardo López, Hiram H. Matthews, Gretchen L. Sarmiento, Eliseo |
| contents | We prove that families of polar codes with multiple kernels over certain symmetric channels can be viewed as polar decreasing monomial-Cartesian codes, offering a unified treatment for such codes, over any finite field. We define decreasing monomial-Cartesian codes as the evaluation of a set of monomials closed under divisibility over a Cartesian product. Polar decreasing monomial-Cartesian codes are decreasing monomial-Cartesian codes whose sets of monomials are closed respect a partial order inspired by the recent work of Bardet, Dragoi, Otmani, and Tillich ["Algebraic properties of polar codes from a new polynomial formalism," 2016 IEEE International Symposium on Information Theory (ISIT)]. Extending the main theorem of Mori and Tanaka ["Source and Channel Polarization Over Finite Fields and Reed-Solomon Matrices," in IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 2720--2736, May 2014], we prove that any sequence of invertible matrices over an arbitrary field satisfying certain conditions polarizes any symmetric over the field channel. In addition, we prove that the dual of a decreasing monomial-Cartesian code is monomially equivalent to a decreasing monomial-Cartesian code. Defining the minimal generating set for a set of monomials, we use it to describe the length, dimension and minimum distance of a decreasing monomial-Cartesian code. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_00325 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Polar decreasing monomial-Cartesian codes Camps, Eduardo López, Hiram H. Matthews, Gretchen L. Sarmiento, Eliseo Information Theory 11T71, 14G50 We prove that families of polar codes with multiple kernels over certain symmetric channels can be viewed as polar decreasing monomial-Cartesian codes, offering a unified treatment for such codes, over any finite field. We define decreasing monomial-Cartesian codes as the evaluation of a set of monomials closed under divisibility over a Cartesian product. Polar decreasing monomial-Cartesian codes are decreasing monomial-Cartesian codes whose sets of monomials are closed respect a partial order inspired by the recent work of Bardet, Dragoi, Otmani, and Tillich ["Algebraic properties of polar codes from a new polynomial formalism," 2016 IEEE International Symposium on Information Theory (ISIT)]. Extending the main theorem of Mori and Tanaka ["Source and Channel Polarization Over Finite Fields and Reed-Solomon Matrices," in IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 2720--2736, May 2014], we prove that any sequence of invertible matrices over an arbitrary field satisfying certain conditions polarizes any symmetric over the field channel. In addition, we prove that the dual of a decreasing monomial-Cartesian code is monomially equivalent to a decreasing monomial-Cartesian code. Defining the minimal generating set for a set of monomials, we use it to describe the length, dimension and minimum distance of a decreasing monomial-Cartesian code. |
| title | Polar decreasing monomial-Cartesian codes |
| topic | Information Theory 11T71, 14G50 |
| url | https://arxiv.org/abs/2002.00325 |