12 papers · ranked by Valyu relevance
Vlad-Florin Drăgoi, Gabriela Cristescu, Dimitrie Popescu
Monomial codes were recently equipped with partial order relations, a fact that allowed researchers to discover structural properties and efficient algorithm for constructing polar codes. Here, we refine the existing order relations in the particular case of the binary erasure channel. The new order relation takes us…
Jicheng Ma, Guiying Yan
Over the past years, Polar codes have arisen as a highly effective class of linear codes, equipped with a decoding algorithm of low computational complexity. This family of codes share a common algebraic formalism with the well-known Reed-Muller codes, which involves monomial evaluations. As useful algebraic codes…
Mohammad Rowshan, Vlad Drăgoi, Jinhong Yuan
—The weight distribution of error correction codes is a critical determinant of their error-correcting performance, making enumeration of utmost importance. In the case of polar codes, the minimum weight wmin (which is equal to minimum distance d) is the only weight for which an explicit enumerator formula is currently…
El Mahdi Mouloua, Mustapha Najmeddine, María Isabel García Planas, Hassan Ou-azzou
'Hassan Ou-azzou'] > Abstract. The monomial codes over a Galois field Fq that can be thought invariant subspaces are essential to us in this study. More specifically, we look into the link between monomial codes and characteristic subspaces and the decomposition of monomial codes into minimal invariant subspaces.…
Eduardo Camps-Moreno, Hiram H. López, Gretchen L. Matthews, Eliseo Sarmiento
'Eliseo Sarmiento'] Abstract. 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…
Hiram H. López, Ivan Soprunov, Rafael H. Villarreal
The aim of this work is to study the dual and the algebraic dual of an evaluation code using standard monomials and indicator functions. We show that the dual of an evaluation code is the evaluation code of the algebraic dual. We develop an algorithm for computing a basis for the algebraic dual. Let C1 and C2 be linear…
Hiram H. López, Gretchen L. Matthews, Ivan Soprunov
A monomial-Cartesian code is an evaluation code defined by evaluating a set of monomials over a Cartesian product. It is a generalization of some families of codes in the literature, for instance toric codes, affine Cartesian codes and J -affine variety codes. In this work we use the vanishing ideal of the Cartesian…
Eduardo Camps-Moreno, Hiram Lopez, Gretchen L. Matthews, Narayanan Rengaswamy + 1 more
In this paper, we focus on the problem of computing the set of diagonal transversal gates fixing a CSS code. We determine the logical actions of the gates as well as the groups of transversal gates that induce non-trivial logical gates and logical identities. We explicitly declare the set of equations defining the…
Keer Wei, Xiaoyu Jin, Weihua Yang, Jun Chen + 1 more
Over binary-input memoryless symmetric (BMS) channels, the performance of polar codes under successive cancellation list (SCL) decoding can approach maximum likelihood (ML) algorithm when the list size L is greater than or equal to $2MF$, where MF, known as mixing factor of code, represents the number of information…
Beatriz Barbero-Lucas, Fernando Hernando, Helena Martín-Cruz, Gary McGuire
'Gary McGuire'] We construct new stabilizer quantum error-correcting codes from generalized monomial-Cartesian codes. Our construction uses an explicitly defined twist vector, and we present formulas for the minimum distance and dimension. Generalized monomial-Cartesian codes arise from polynomials in m variables. When…
Pranjal Dutta, Mahesh Sreekumar Rajasree, Santanu Sarkar
\usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\text{ TinyJAMBU }$$\end{document} TinyJAMBU Authors: ['Pranjal Dutta' 'Mahesh Sreekumar Rajasree' 'Santanu Sarkar'] In…
Jan Schoone, Joan Daemen
\usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\chi _n$$\end{document} χ n Authors: ['Jan Schoone' 'Joan Daemen'] The Boolean map $\chi _n :\mathbb{F}_2^n \rightarrow…