arXiv subjects

Gustavo Terra Bastos

Publications and source records attributed to Gustavo Terra Bastos.

7 recordsLinked to original sources

arXiv2026

Decoding Algorithms for MDS Array Codes

We study decoding procedures for a family of MDS array codes previously constructed from the Kronecker product of a superregular matrix and a non-singular matrix over a finite field. By exploiting the particular structure of their parity-check matrices, we develop decoding algorithms for different channel models. For the erasure channel, we provide an algorithm capable of recovering any pattern of up to $n-k$ symbol erasures. For the $q$-ary symmetric channel, we investigate the decoding of one and two symbol errors and give explicit procedures for determining their locations and values. We also consider the particular case in which the superregular matrix is a Vandermonde matrix, showing how its additional algebraic structure can be exploited in the decoding process. Explicit examples over different finite fields are provided to illustrate the proposed procedures.

cs.IT↗

arXiv2026

A Construction of Infinite Families of Good Self-Orthogonal Quasi-Cyclic Codes

Quasi-cyclic codes have been recently employed in the constructions of quantum error-correcting codes. In this paper, we propose a construction of infinite families of quasi-cyclic codes over $\F_q$ which are self-orthogonal with respect to the Euclidean and Hermitian inner products. In particular, their dimension and a lower bound for their minimum distance are computed using their constituent codes defined over field extensions of $\mathbb{F}_q$. We also show that the lower bound for the minimum distance satisfies the square-root-like lower bound and also show how dual-containing and self-dual quasi-cyclic codes can arise from our construction. Using the CSS construction, we show the existence of quantum error-correcting codes with good parameters.

cs.IT↗

arXiv2025

Constructing Superregular Matrices

Superregular matrices, i.e., matrices where all square submatrices are non-singular, have a wide range of applications in communications. A superregular block matrix is a broader concept where all full block submatrices, with the appropriate size, are non-singular. In this work we propose a construction of block superregular matrices based on the Kronecker product of superregular matrices with non-singular matrices. Furthermore, we propose two constructions of superregular matrices via other smaller superregular matrices over smaller fields.

math.RA↗

arXiv2024

A Construction of Optimal Quasi-cyclic Locally Recoverable Codes using Constituent Codes

A locally recoverable code of locality $r$ over $\mathbb{F}_{q}$ is a code where every coordinate of a codeword can be recovered using the values of at most $r$ other coordinates of that codeword. Locally recoverable codes are efficient at restoring corrupted messages and data which make them highly applicable to distributed storage systems. Quasi-cyclic codes of length $n=m\ell$ and index $\ell$ are linear codes that are invariant under cyclic shifts by $\ell$ places. %Quasi-cyclic codes are generalizations of cyclic codes and are isomorphic to $\mathbb{F}_{q} [x]/ \langle x^m-1 \rangle$-submodules of $\mathbb{F}_{q^\ell} [x] / \langle x^m-1 \rangle$. In this paper, we decompose quasi-cyclic locally recoverable codes into a sum of constituent codes where each constituent code is a linear code over a field extension of $\mathbb{F}_q$. Using these constituent codes with set parameters, we propose conditions which ensure the existence of almost optimal and optimal quasi-cyclic locally recoverable codes with increased dimension and code length.

cs.IT↗

arXiv2023

New results on linear permutation polynomials with coefficients in a subfield

Some families of linear permutation polynomials of $\mathbb{F}_{q^{ms}}$ with coefficients in $\mathbb{F}_{q^{m}}$ are explicitly described (via conditions on their coefficients) as isomorphic images of classical subgroups of the general linear group of degree $m$ over the ring $\frac{\mathbb{F}_{q}[x]}{\left\langle x^{s}-1 \right\rangle}$. In addition, the sizes of some of these families are computed. Finally, several criteria to construct linear permutation polynomials of $\mathbb{F}_{q^{2p}}$ (where $p$ is a prime number) with prescribed coefficients in $\mathbb{F}_{q^{2}}$ are given. Examples illustrating the main results are presented.

math.RT↗

arXiv2020

Linear Permutations and their Compositional Inverses over $\mathbb{F}_{q^n}$

The use of permutation polynomials has appeared, along to their compositional inverses, as a good choice in the implementation of cryptographic systems. Hence, there has been a demand for constructions of these polynomials which coefficients belong to a finite field. As a particular case of permutation polynomial, involution is highly desired since its compositional inverse is itself. In this work, we present an effective way of how to construct several linear permutation polynomials over $\mathbb{F}_{q^n}$ as well as their compositional inverses using a decomposition of $\displaystyle{\frac{\mathbb{F}_q[x]}{\left\langle x^n -1 \right\rangle}}$ based on its primitive idempotents. As a consequence, an immediate construction of involutions is presented.

math.NT↗

arXiv2018

Abelian Noncyclic Orbit Codes and Multishot Subspace Codes

In this paper we characterize the orbit codes as geometrically uniform codes. This characterization is based on the description of all isometries over a projective geometry. In addition, the Abelian orbit codes are defined and a new construction of Abelian non-cyclic orbit codes is presented. In order to analyze their structures, the concept of geometrically uniform partitions have to be reinterpreted. As a consequence, a substantial reduction in the number of computations needed to obtain the minimum subspace distance of these codes is achieved and established. An application of orbit codes to multishot subspace codes obtained according to a multi-level construction is provided.

cs.IT↗