A decreasing Cartesian code is defined by evaluating a monomial set closed under divisibility on a Cartesian set. Some well-known examples are the Reed-Solomon, Reed-Muller, and (some) toric codes. The affine permutations consist of the permutations of the code that depend on an affine transformation. In this work, we study the affine permutations of some decreasing Cartesian codes, including the case when the Cartesian set has copies of multiplicative or additive subgroups.

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.

In this paper we show some applications of algebraic curves to the construction of kernels of polar codes over a discrete memoryless channel which is symmetric w.r.t the field operations. We will also study the minimum distance of the polar codes proposed, their duals and the exponents of the matrices used for defining them. All the restrictions that we make to our curves will be accomplished by the so called Castle Curves.