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.

Abstract The manipulation of 3D objects is becoming crucial for many applications, such as health, industry, or entertainment, to mention some. However, these 3D objects require substantial energy and different types of resources. With the goal of obtaining a simplified representation of a 3D object that can be easily managed, for example, for transmission, in some recent works, the authors associate low-density point clouds with a 3D object that simplifies the original 3D object. More precisely, given a 3D object in a polyhedral format, some authors associate a chain code and then use grammar-free context to obtain key points that give rise to several point clouds with different densities. In this work, we complete the cycle by developing a polyhedral reconstruction from an associated low-density point cloud and the chain code. The polyhedral reconstruction is crucial for handling 3D objects because it allows us to visualize them after they are efficiently compressed and transmitted. We apply our algorithms to well-known 3D objects in the literature. We use the Hausdorff and Chamfer distances to compare our results with the state-of-the-art proposals. We show how our proposed polyhedral reconstruction based on a helical chain code reconstructs a medical image represented or transmitted by slices into a 3D object in a polyhedral format, helping thus to mitigate and alleviate the management of 3D medical objects. The polyhedron that we propose provides better compression when compared with the original set of slices of a 3D medical object.

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.

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.

As the field of quantum computing rapidly advances, there is a growing demand for skilled professionals adept in quantum computing and programming. Recognizing this need, in this paper, we share our experiences teaching an introductory-level quantum computing course to students at Cleveland State University (CSU). The course integrates dedicated hands-on programming labs, allowing students to verify their experimental results with corresponding examples from the textbook. These labs cover a diverse range of topics, including fundamental elements such as quantum gates and circuits, quantum key distribution protocols, and quantum algorithms. As educators, our goal is to share teaching insights and resources with fellow instructors in the field. This article elucidates the rationale behind the design of each experiment, providing a deeper understanding of quantum computing.

In this paper, we introduce multivariate Goppa codes, which contain, as a particular case, the well-known classical Goppa codes. We provide a parity check matrix for a multivariate Goppa code in terms of a tensor product of generalized Reed-Solomon codes. We prove that multivariate Goppa codes are subfield subcodes of augmented Cartesian codes. By showing how this new family of codes relates to a tensor product of generalized Reed-Solomon codes and augmented codes, we obtain information about the parameters, subcodes, duals, and hulls of multivariate Goppa codes. We see that in some instances, the hulls of multivariate Goppa codes (resp., tensor product of generalized Reed-Solomon codes) are also multivariate Goppa codes (resp. tensor product of generalized Reed-Solomon codes). We utilize the multivariate Goppa codes to obtain entanglement-assisted quantum error-correcting codes and to build families of long LCD, self-dual, or self-orthogonal codes.

This paper presents secure MatDot codes, a family of evaluation codes that support secure distributed matrix multiplication via a careful selection of evaluation points that exploit the properties of the dual code. We show that the secure MatDot codes provide security against the user by using locally recoverable codes. These new codes complement the recently studied discrete Fourier transform codes for distributed matrix multiplication schemes that also provide security against the user. There are scenarios where the associated costs are the same for both families and instances where the secure MatDot codes offer a lower cost. In addition, the secure MatDot code provides an alternative way to handle the matrix multiplication by identifying the fastest servers in advance. In this way, it can determine a product using fewer servers, specified in advance, than the MatDot codes which achieve the optimal recovery threshold for distributed matrix multiplication schemes.

In this work, we present linear exact repair schemes for one or two erasures in decreasing monomial-Cartesian codes (DM-CC), a family of codes which provides a framework for polar codes. In the case of two erasures, the positions of the erasures should satisfy a certain restriction. We present families of augmented Reed-Muller (ARM) and augmented Cartesian codes (ACar) which are families of evaluation codes obtained by strategically adding vectors to Reed-Muller and Cartesian codes, respectively. We develop repair schemes for one or two erasures for these families of augmented codes. Unlike the repair scheme for two erasures of DM-CC, the repair scheme for two erasures for the augmented codes has no restrictions on the positions of the erasures. When the dimension and base field are fixed, we give examples where ARM and ACar codes provide a lower bandwidth (resp., bitwidth) in comparison with Reed-Solomon (resp., Hermitian) codes. When the length and base field are fixed, we give examples where ACar codes provide a lower bandwidth in comparison with ARM. Finally, we analyze the asymptotic behavior when the augmented codes achieve the maximum rate.

We present two families of augmented Reed-Muller (ARM) codes, which are evaluation codes obtained by adding specific vectors to a Reed-Muller code. We develop exact repair schemes for single erasures for these ARM codes. When a dimension and a base field are fixed, we give examples where ARM codes provide a lower bandwidth in comparison with Reed-Solomon codes. We analyze the asymptotical behavior when ARM codes achieve the maximum rate.

In this paper, we construct codes for local recovery of erasures with high availability and constant-bounded rate from the Hermitian curve. These new codes, called Hermitian-lifted codes, are evaluation codes with evaluation set being the set of $\mathbb{F}_{q^2}$-rational points on the affine curve. The novelty is in terms of the functions to be evaluated; they are a special set of monomials which restrict to low degree polynomials on lines intersected with the Hermitian curve. As a result, the positions corresponding to points on any line through a given point act as a recovery set for the position corresponding to that point.

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.