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E. J. García-Claro

Publications and source records attributed to E. J. García-Claro.

4 recordsLinked to original sources
arXiv2025

On reversible and reversible-complementary DNA codes over $\mathbb{F}_{4}$

A method to construct and count all the linear codes (of arbitrary length) in $\mathbb{F}_{4}$ that are invariant under reverse permutation and that contain the repetition code is presented. These codes are suitable for constructing DNA codes that satisfy the reverse and reverse-complement constraints. By analyzing a module-theoretic structure of these codes, their generating matrices are characterized in terms of their isomorphism type, and explicit formulas for counting them are provided. The proposed construction method based on this characterization outperforms the one given by Abualrub et al. for cyclic codes (of odd length) over $\mathbb{F}_{4}$, and the counting method solves a problem that can not be solved using the one given by Fripertinger for invariant subspaces under a linear endomorphism of $\mathbb{F}_{q}^{n}$. Additionally, several upper bounds and an identity for the minimum Hamming distance of certain reversible codes are provided.

cs.IT↗
arXiv2024

On Grid Codes

Generating functions for the size of a $r$-sphere, with respect to the Manhattan distance in an $n$-dimensional grid, are used to provide explicit formulas for the minimum and maximum size of an $r$-ball centered at a point of the grid. This allows us to offer versions of the Hamming and Gilbert-Varshamov bounds for codes in these grids. Relations between the Hamming, Manhattan, and Lee distances defined in an abelian group $G$ are studied. A formula for the minimum Hamming distance of codes that are cyclic subgroups of $G$ is presented. Furthermore, several lower bounds for the minimum Manhattan distance of these codes based on their minimum Hamming and Lee distances are established. Examples illustrating the main results are presented, including several SageMath implementations.

cs.IT↗
arXiv2024

On easily computable indecomposable dimension group algebras, and group codes

An easily computable dimension (or ECD) group code in the group algebra $\mathbb{F}_{q}G$ is an ideal of dimension less than or equal to $p=char(\mathbb{F}_{q})$ that is generated by an idempotent. This paper introduces an easily computable indecomposable dimension (or ECID) group algebra as a finite group algebra for which all group codes generated by primitive idempotents are ECD. Several characterizations are given for these algebras. In addition, some arithmetic conditions to determine whether a group algebra is ECID are presented, in the case it is semisimple. In the non-semisimple case, these algebras have finite representation type where the Sylow $p$-subgroups of the underlying group are simple. The dimension and some lower bounds for the minimum Hamming distance of group codes in these algebras are given together with some arithmetical tests of primitivity of idempotents. Examples illustrating the main results are presented.

math.RT↗