Ismael Gutiérrez | Searcharxiv

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↗

arXiv2014

Multiplicative Structures on the Twisted Equivariant K-theory of Finite Groups

Let $K$ be a finite group and let $G$ be a finite group acting on $K$ by automorphisms. In this paper we study two different but intimately related subjects: on the one side we classify all possible multiplicative and associative structures with which one can endow the twisted $G$-equivariant K-theory of $K$, and on the other, we classify all possible monoidal structures with which one can endow the category of twisted and $G$-equivariant bundles over $K$. We achieve this classification by encoding the relevant information in the cochains of a sub double complex of the double bar resolution associated to the semi-direct product $K \rtimes G$; we use known calculations of the cohomology of $K$, $G$ and $K \rtimes G$ to produce concrete examples of our classification. In the case in which $K=G$ and $G$ acts by conjugation, the multiplication map $G \rtimes G \to G$ is a homomorphism of groups and we define a shuffle homomorphism which realizes this map at the homological level. We show that the categorical information that defines the Twisted Drinfeld Double can be realized as the dual of the shuffle homomorphism applied to any 3-cocycle of $G$. We use the pullback of the multiplication map in cohomology to classify the possible ring structures that the Grothendieck ring of representations of the Twisted Drinfeld Double may have, and we include concrete examples of this procedure.

math.QA↗