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Javier de la Cruz

Publications and source records attributed to Javier de la Cruz.

At least 19 recordsLinked to original sources
arXiv2026

The Neighbor Graph of Linear Complementary Dual (LCD) Codes

Linear complementary dual (LCD) codes form an important class of linear codes with applications in cryptography, classical error correction, and quantum coding theory. In this paper, we study the neighbor relation on LCD codes over finite fields and the graph induced by this relation, where two codes are adjacent whenever they intersect in codimension one. We determine the number of neighbors of an LCD code that are also LCD, and we use this result to analyze the structure of the corresponding neighbor graph. In particular, we prove its regularity over arbitrary finite fields and establish further regularity properties for its main structural subgraphs in the binary and odd-characteristic cases. These results provide a graph-theoretic framework for the study of LCD codes and reveal a strong combinatorial regularity in their neighborhood structure.

math.CO↗
arXiv2026

Projector additive group codes

Let $F=\mathbb{F}_q$ and let $K=\mathbb{F}_{q^m}$ be a finite extension of $F$. An additive left group code is a left $FG$-submodule of the group algebra $KG$. Classical idempotent group codes describe images of $KG$-linear projectors and are necessarily $K$-linear. They therefore do not provide a sufficiently broad framework for additive group codes, which need not be $K$-linear. In this paper, we develop a projector-based approach for the study of additive left group codes. We distinguish between restricted idempotent additive group codes of the form $FGe$, where $e\in KG$ is an idempotent, and projector additive left group codes, which are images of arbitrary $FG$-linear projectors on $KG$. While $KG$-linear projector group codes coincide with classical idempotent group codes, their additive counterparts do not, in general, coincide. We show that projector additive group codes provide constructions beyond the restricted idempotent setting, and we give examples in both the semisimple and non-semisimple cases with parameters not attained by restricted idempotent additive group codes. The projector framework also provides effective algebraic tools for studying duality. We relate trace-Euclidean and trace-Hermitian duality to adjoints of $FG$-linear projectors, characterize LCD additive group codes through self-adjoint projectors, and obtain sufficient conditions for self-duality. We further study Murray--von Neumann equivalence of projectors and show that it characterizes isomorphism of their images as left $FG$-modules, thereby providing a module-theoretic classification tool for projector additive group codes. Finally, we interpret quotients by orthogonal codes in terms of module duals.

math.RA↗
arXiv2026

Twisted group algebras of faithful split metacyclic groups $C_p \rtimes C_m$ over finite fields

Let $\mathbb{F}_\ell$ be a finite field with $\ell$ elements and let $G = C_p \rtimes C_m$ be a faithful split metacyclic group. In this paper, we develop a complete theory for the twisted group algebra $\mathbb{F}_\ell^αG$. Using the Lyndon--Hochschild--Serre spectral sequence, we prove that the second cohomology group of $G$ is isomorphic to $\mathbb{F}_\ell^\times/(\mathbb{F}_\ell^\times)^m$, and we show that all twisting occurs only on the $C_m$ factor. We determine the primitive central idempotents by analyzing the combined action of the Frobenius automorphism and the group action on the character group of $C_p$. Using crossed product theory and the structure of finite fields, we obtain the complete Wedderburn decomposition of $\mathbb{F}_\ell^αG$ into matrix algebras over explicitly determined fields $\mathbb{F}_{\ell^{d_j}}$. Finally, the irreducible projective representations of $G$ over $\mathbb{F}_\ell$ are also determined.

math.RA↗
arXiv2025

Duality on group algebras over finite chain rings: applications to additive group codes

Given a finite group $G$ and an extension of finite chain rings $S|R$, one can consider the group rings $\mathscr{S} = S[G]$ and $\mathscr{R} = R[G]$. The group ring $\mathscr{S}$ can be viewed as an $R$-bimodule, and any of its $R$-submodules naturally inherits an $R$-bimodule structure; in the framework of coding theory, these are called \emph{additive group codes}, more precisely a (left) additive group code of is a linear code which is the image of a (left) ideal of a group algebra via an isomorphism which maps $G$ to the standard basis of $S^n$, where $n=|G|$. In the first part of the paper, the ring extension $S|R$ is studied, and several $R$-module isomorphisms are established for decomposing group rings, thereby providing a characterization of the structure of additive group codes. In the second part, we construct a symmetric, nondegenerate trace-Euclidean inner product on $\mathscr{S}$. Two additive group codes $\mathcal{C}$ and $\mathcal{D}$ form an \emph{additive complementary pair} (ACP) if $\mathcal{C} + \mathcal{D} = \mathscr{S}$ and $\mathcal{C} \cap \mathcal{D} = \{0\}$. For two-sided ACPs, we prove that the orthogonal complement of one code under the trace-Euclidean duality is precisely the image of the other under an involutive anti-automorphism of $\mathscr{S}$, linking coding-theoretical ACPs with module orthogonal direct-sum decompositions, representation theory, and the structure of group algebras over finite chain rings.

cs.IT↗
arXiv2025

Diagnosing Thermalization via Participation Ratio in Disordered Bosonic Chains

We study thermalization in a disordered one-dimensional interacting bosonic system described by the Aubry-Andre model using full exact diagonalization. We find a broad chaotic energy window where the system's eigenstates satisfy the Eigenstate Thermalization Hypothesis (ETH), demonstrated by the smooth energy dependence of observables like entanglement entropy and local particle number, whose fluctuations decrease with system size. Dynamically, we investigate the equilibration of initial Fock states and find that thermalization is not universal. The key finding is a direct and nontrivial correlation between an initial state's delocalization in the energy eigenbasis quantified by the Participation Ratio (PR) and its subsequent equilibration. States with a high PR consistently evolve toward the microcanonical ensemble prediction, whereas those exhibiting a low PR display deviations whose magnitude inversely correlates with the PR value. This connection is quantitatively confirmed by the trace distance, providing a powerful, experimentally relevant diagnostic for predicting which initial states will reach thermal equilibrium.

quant-ph↗
arXiv2023

Persistent revivals in a system of trapped bosonic atoms

Dynamical signatures of quantum chaos are observed in the survival probability of different initial states, in a system of cold atoms trapped in a linear chain with site noise and open boundary conditions. It is shown that chaos is present in the region of small disorder, at intermediate energies. The study is performed with different number of sites and atoms: 7,8 and 9, but focusing on the case where the particle density is one. States of the occupation basis with energies in the chaotic region are evolved at long times. Remarkable differences in the behaviour of the survival probability are found for states with different energy-eigenbasis participation ratio (PR). Whereas those with large PR clearly exhibit the characteristic random-matrix correlation hole before equilibration, those with small PR present a marginal or even no correlation hole which is replaced by revivals lasting up to the stage of equilibration, suggesting a connection with the quantum scarring phenomenon.

quant-ph↗
arXiv2023

On LCP codes over a mixed ring alphabet

In this paper, we introduce a standard generator matrix for mixed-alphabet linear codes over finite chain rings. Furthermore, we show that, when one has a linear complementary pair (LCP) of mixed-alphabet linear codes, both codes are weakly-free. Additionally, we establish that any mixed-alphabet product group code is separable. Thus, if one has a pair $\{C, D\}$ of mixed-alphabet product group codes over a finite chain ring that forms a LCP, it follows that $C$ and the Euclidean dual of $D$ are permutation equivalent.

cs.IT↗
arXiv2022

Twisted skew $G$-codes

In this paper we investigate left ideals as codes in twisted skew group rings. The considered rings, which are often algebras over a finite field, allows us to detect many of the well-known codes. The presentation, given here, unifies the concept of group codes, twisted group codes and skew group codes.

cs.IT↗
arXiv2022

Public key cryptography based on skew dihedral group rings

In this paper, we propose to use a skew dihedral group ring given by the group $D_{2n}$ and the finite field $\mathbb{F}_{q^2}$ for public-key cryptography. Using the ambient space $\mathbb{F}_{q^{2}}^θ D_{2n}$ and a group homomorphism $θ: D_{2n} \rightarrow \mathrm{Aut}(\mathbb{F}_{q^2})$, we introduce a key exchange protocol and present an analysis of its security. Moreover, we explore the properties of the resulting skew group ring $\mathbb{F}_{q^{2}}^θ D_{2n}$, exploiting them to enhance our key exchange protocol. We also introduce a probabilistic public-key scheme derived from our key exchange protocol and obtain a key encapsulation mechanism (KEM) by applying a well-known generic transformation to our public-key scheme. Finally, we present a proof-of-concept implementation of our cryptographic constructions. To the best of our knowledge, this is the first paper that proposes a skew dihedral group ring for public-key cryptography.

cs.CR↗
arXiv2021

Public key cryptography based on twisted dihedral group algebras

In this paper, we propose to use a twisted dihedral group algebra for public-key cryptography. For this, we introduce a new $2$-cocycle $α_λ$ to twist the dihedral group algebra. Using the ambient space $\mathbb{F}^{α_λ} D_{2n}$, we then introduce a key exchange protocol and present an analysis of its security. Moreover, we explore the properties of the resulting twisted algebra $\mathbb{F}^{α_λ}D_{2n}$, exploiting them to enhance our key exchange protocol. We also introduce a probabilistic public-key scheme derived from our key-exchange protocol and obtain a key encapsulation mechanism (KEM) by applying a well-known generic transformation to our public-key scheme. Finally, we present a proof-of-concept implementation of the resulting key encapsulation mechanism.

cs.CR↗
arXiv2021

On checkable codes in group algebras

We classify, in terms of the structure of the finite group G, all group algebras KG for which all right ideals are right annihilators of principal left ideals. This means in the language of coding theory that we classify code-checkable group algebras KG which have been considered so far only for abelian groups G. Optimality of checkable codes and asymptotic results are discussed.

cs.IT↗
arXiv2020

Quantum chaos in a system with high degree of symmetries

We study dynamical signatures of quantum chaos in one of the most relevant models in many-body quantum mechanics, the Bose-Hubbard model, whose high degree of symmetries yields a large number of invariant subspaces and degenerate energy levels. While the standard procedure to reveal signatures of quantum chaos requires classifying the energy levels according to their symmetries, we show that this classification is not necessary to obtain manifestation of spectral correlations in the temporal evolution of the survival probability. Our findings exhibit the survival probability as a powerful tool to detect the presence of quantum chaos, avoiding the experimental and theoretical challenges associated with the determination of a complete set of energy eigenstates and their symmetry classification.

quant-ph↗
arXiv2019

A Note on Linear Complementary Pairs of Group Codes

We give a short and elementary proof of the fact that for a linear complementary pair $(C,D)$, where $C$ and $D$ are $2$-sided ideals in a group algebra, $D$ is uniquely determined by $C$ and the dual code $D^\perp$ is permutation equivalent to $C$. This includes earlier results of Carlet et al. and Güneri et al. on nD cyclic codes which have been proved by subtle and lengthy calculations in the space of polynomials.

cs.IT↗
arXiv2017

Some new results on the self-dual [120,60,24] code

The existence of an extremal self-dual binary linear code of length 120 is a long-standing open problem. We continue the investigation of its automorphism group, proving that automorphisms of order 30 and 57 cannot occur. Supposing the involutions acting fixed point freely, we show that also automorphisms of order 8 cannot occur and the automorphism group is of order at most 120, with further restrictions. Finally, we present some necessary conditions for the existence of the code, based on shadow and design theory.

cs.IT↗
arXiv2016

On dually almost MRD codes

In this paper we define and study a family of codes which come close to be MRD codes, so we call them AMRD codes (almost MRD). An AMRD code is a code with rank defect equal to 1. AMRD codes whose duals are AMRD are called dually AMRD. Dually AMRD codes are the closest to the MRD codes given that both they and their dual codes are almost optimal. Necessary and sufficient conditions for the codes to be dually AMRD are given. Furthermore we show that dually AMRD codes and codes of rank defect one and maximum 2-generalized weight coincide when the size of the matrix divides the dimension.

cs.IT↗