In this work, we prove that the permutation group of a Reed-Solomon code is given by the polynomials of degree one that leave the set of evaluation points invariant. Our results provide a straightforward proof of the well-known cases of the permutation group of the Reed-Solomon code when the set of evaluation points is the whole finite field or the multiplicative group.

Abstract The Generalized Hamming weights and their relative version, which generalize the minimum distance of a linear code, are relevant to numerous applications, including coding on the wire-tap channel of type II, t-resilient functions, bounding the cardinality of the output in list decoding algorithms, ramp secret sharing schemes, and quantum error correction. The generalized Hamming weights have been determined for some families of codes, including Cartesian codes and Hermitian one-point codes. In this paper, we determine the generalized Hamming weights of decreasing norm-trace codes, which are linear codes defined by evaluating sets of monomials that are closed under divisibility on the rational points of the extended norm-trace curve given by $$x^{u} = y^{q^{s - 1}} + y^{q^{s - 2}} + \cdots + y$$ x u = y q s - 1 + y q s - 2 + ⋯ + y over the finite field of cardinality $$q^s$$ q s , where u is a positive divisor of $$\frac{q^s - 1}{q - 1}$$ q s - 1 q - 1 . As a particular case, we obtain the weight hierarchy of one-point norm-trace codes and recover the result of Barbero and Munuera (2001) giving the weight hierarchy of one-point Hermitian codes. We also study the relative generalized Hamming weights for these codes and use them to construct impure quantum codes with excellent parameters.

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 paper, we consider using norm-trace codes over extension fields for error correction using partial information from received words. To do so, we define virtual projections of norm-trace codes and we implement a fractional decoding scheme. The scheme depends on a refined key equation tailored to the norm-trace code.

Given two$q$-ary codes$C_{1}$and$C_{2}$, the relative hull of$C_{1}$with respect to$C_{2}$is the intersection$C_{1}\cap C_{2}^{\perp} $. We prove that when$q>2$, the relative hull dimension can be repeatedly reduced by one, down to a certain bound, by replacing either of the two codes with an equivalent one. The reduction of the relative hull dimension applies to hulls taken with respect to the$e$-Galois inner product, which has as special cases both the Euclidean and Hermitian inner products. We give conditions under which the relative hull dimension can be increased by one via equivalent codes when$q>2$. We study some consequences of the relative hull properties on entanglement-assisted quantum error-correcting codes and prove the existence of new entanglement-assisted quantum error-correcting maximum distance separable codes, meaning those whose parameters satisfy the quantum Singleton bound.

Sum-rank metric codes have recently attracted the attention of many researchers, due to their relevance in several applications. Mathematically, the sum-rank metric is a natural generalization of both the Hamming metric and the rank metric. In this paper, we provide an Anticode Bound for the sum-rank metric, which extends the corresponding Hamming and rank-metric Anticode bounds. We classify then optimal anticodes, i.e., codes attaining the sum-rank metric Anticode Bound. We use these optimal anticodes to define generalized sum-rank weights and we study their main properties. In particular, we prove that the generalized weights of an MSRD code are determined by its parameters. As an application, in the Appendix we explain how generalized weights measure information leakage in multishot network coding.

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.