arXiv subjects

Antonio Aparecido de Andrade

Publications and source records attributed to Antonio Aparecido de Andrade.

8 recordsLinked to original sources

arXiv2026

On The Cyclicity of Algebraic Lattices

This work presents theoretical advances in the study of cyclic and quasi-cyclic lattices. First, we provide elementary facts regarding cyclic and quasi-cyclic lattices, and discuss about the cyclicity of some notable lattices. The main contributions of the paper is in the algebraic setting: we investigate cyclic lattices arising from $\mathbb{Z}$-modules in Galois number fields via the Minkowski embedding. We establish necessary and sufficient conditions for an algebraic lattice to be cyclic over both cyclic and general Galois number fields, expressed in terms of naturally associated groups. Moreover, we derive a necessary and sufficient condition for ideal lattices to be cyclic, depending on the factorization of the ideal in the underlying number field.

math.NT↗

arXiv2026

Algebraic Lattices Arising from Congruence Submodules in Subfields of $p$-th Cyclotomic Fields

The classical sphere packing problem, which remains unsolved, consists of determining how densely a large number of identical spheres can be packed together. In some sphere packings, the centers of the spheres in a sphere packing form a Euclidean lattice, which is a discrete additive subgroup of $\mathbb{R}^n$. Free $\mathbb{Z}$-modules in the ring of integers of an algebraic number field yield algebraic lattices via the canonical embedding. In this work, we present new constructions of algebraic lattices from certain families of $\mathbb{Z}$-modules in the ring of algebraic integers of subfields of the $p$-th cyclotomic field, where $p$ is a prime number. Within this framework, we compute lower bounds for the center density of these algebraic lattices and construct algebraic lattices having the best known packing density in dimensions 2, 3, and 5.

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arXiv2026

Non-norm criteria and optimal $2\times 2$ space-time block codes over rings of integers of imaginary quadratic fields

Codes arising from algebraic structures over number fields lead naturally to determinant optimization problems governed by arithmetic invariants. In this paper, we investigate $2\times 2$ space-time block codes defined over rings of integers of imaginary quadratic fields, combining tools from algebraic number theory, cyclic algebras, and lattice theory. We prove that the Eisenstein construction over $\mathbb{Z}[ζ_3]$ is optimal within the family considered here: it attains the largest normalized density among the $2\times 2$ space-time block codes arising from rings of integers of imaginary quadratic fields. As a first step, we show that any code that could improve upon the Eisenstein construction must be defined over the ring of integers of $\mathbb{Q}(\sqrt{-d})$ with $d\in\{2,7,11\}$, apart from the classical Gaussian and Eisenstein cases. We then analyze these remaining fields by explicit arithmetic arguments, determine the optimal constructions over them, and show that none of them improves upon the Eisenstein code. A key ingredient in our approach is the derivation of effective non-norm criteria for quadratic extensions of imaginary quadratic fields. These criteria are obtained by local methods involving $2$-adic and $3$-adic valuations together with Hensel's lemma, and they ensure the division algebra property required for full diversity. They may also be of independent interest in the study of division algebras and their applications to coding theory and lattice-based communication.

cs.IT↗

arXiv2025

Constructions of well-rounded algebraic lattices over odd prime degree cyclic number fields

Algebraic lattices are those obtained from modules in the ring of integers of algebraic number fields through the canonical or twisted embeddings. In turn, well-rounded lattices are those with maximal cardinality of linearly independent vectors in its set of minimal vectors. Both classes of lattices have been applied for signal transmission in some channels, such as wiretap channels. Recently, some advances have been made in the search for well-rounded lattices that can be realized as algebraic lattices. Moreover, some works have been published studying algebraic lattices obtained from modules in cyclic number fields of odd prime degree $p$. In this work, we generalize some results of a recent work of Tran et al. and we provide new constructions of well-rounded algebraic lattices from a certain family of modules in the ring of integers of each of these fields when $p$ is ramified in its extension over the field of rational numbers.

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arXiv2024

Characterization of $n$-Dimensional Toric and Burst-Error-Correcting Quantum Codes from Lattice Codes

Quantum error correction is essential for the development of any scalable quantum computer. In this work we introduce a generalization of a quantum interleaving method for combating clusters of errors in toric quantum error-correcting codes. We present new $n$-dimensional toric quantum codes, where $n\geq 5$, which are featured by lattice codes and apply the proposed quantum interleaving method to such new $n$-dimensional toric quantum codes. Through the application of this method to these novel $n$-dimensional toric quantum codes we derive new $n$-dimensional quantum burst-error-correcting codes. Consequently, $n$-dimensional toric quantum codes and burst-error-correcting quantum codes are provided offering both a good code rate and a significant coding gain when it comes to toric quantum codes. Another important consequence from the presented $n$-dimensional toric quantum codes is that if the Golomb and Welch conjecture in \cite{perfcodes} regarding the Lee sphere in $n$ dimensions for the respective close packings holds true, then it follows that these $n$-dimensional toric quantum codes are the only possible ones to be obtained from lattice codes. Moreover, such a methodology can be applied for burst error correction in cases involving localized errors, quantum data storage and quantum channels with memory.

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arXiv2022

On the Construction of New Toric Quantum Codes and Quantum Burst-Error Correcting Codes

A toric quantum error-correcting code construction procedure is presented in this work. A new class of an infinite family of toric quantum codes is provided by constructing a classical cyclic code on the square lattice $\mathbb{Z}_{q}\times \mathbb{Z}_{q}$ for all odd integers $q\geq 5$ and, consequently, new toric quantum codes are constructed on such square lattices regardless of whether $q$ can be represented as a sum of two squares. Furthermore this work supplies for each $q$ the polyomino shapes that tessellate the corresponding square lattices and, consequently, tile the lattice $\mathbb{Z}^{2}$. The channel without memory to be considered for these constructed toric quantum codes is symmetric, since the $\mathbb{Z}^{2}$-lattice is autodual. Moreover, we propose a quantum interleaving technique by using the constructed toric quantum codes which shows that the code rate and the coding gain of the interleaved toric quantum codes are better than the code rate and the coding gain of Kitaev's toric quantum codes for $q=2n+1$, where $n\geq 2$, and of an infinite class of Bombin and Martin-Delgado's toric quantum codes. In addition to the proposed quantum interleaving technique improves such parameters, it can be used for burst-error correction in errors which are located, quantum data stored and quantum channels with memory.

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arXiv2019

Well-Rounded Lattices via Polynomials

Well-rounded lattices have been a topic of recent studies with applications in wiretap channels and in cryptography. A lattice of full rank in Euclidean space is called well-rounded if its set of minimal vectors spans the whole space. In this paper, we investigate when lattices coming from polynomials with integer coefficients are well-rounded.

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arXiv2013

Bandwidth reduction in cognitive radio

Due to mushroom development of wireless devices cognitive radio is used to resolve the bandwidth utilization and sacristy problem. The crafty usage of bandwidth in cognitive radio based on error correcting codes is ensured to accomodate un authorized user. This study proposes a transmission model by which a finite sequence of binary cyclic codes constructed by a binary BCH code of length $n=2^{s}-1$, in which all codes have same error correction capability and code rate but sequentially increasing code lengths greater than $n$. Initially all these codes are carrying data of their corresponding primary users. A transmission pattern is planned in the sprit of interweave model deals the transmission parameters; modulation scheme, bandwidth and code rate. Whenever, any of the primary users having mod of transmission, the binary cyclic code, is not using its allocated bandwidth, the user having its data built by binary BCH code enter and exploit the free path as a secondary user. Eventually whenever the primary user with $W$ bandwidth having binary BCH code for its data transmission, change its status as a secondary user, it just requires the bandwidth less than $W$.

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