Polar decreasing monomial-Cartesian codes
Eduardo Camps, Hiram H. López, Gretchen L. Matthews, Eliseo Sarmiento
TL;DR
This work addresses unifying polar codes with multiple kernels over symmetric channels by casting them as polar decreasing monomial-Cartesian codes over arbitrary finite fields. The authors develop a duality framework and a constructive description of code parameters using minimal generating sets, and prove that any sequence of invertible kernels with non-identity standard forms polarizes SOF channels. The results yield a unified algebraic perspective that encompasses Reed-Solomon and Cartesian variants and extend known polarization theorems from Mori–Tanaka to general fields. The framework offers concrete tools for analyzing and designing multikernel polar codes with predictable performance across a broad class of channels.
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 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.