### Session S29 - Theory and Applications of Coding Theory

Monday, July 19, 19:00 ~ 19:25 UTC-3

## $t$-graph of distances of a finitely-generated group and block codes

### Ismael Gutierrez

#### Universidad del Norte, Colombia - This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $G$ be a finitely-generated group with generating set $M=\{g_1,\ldots, g_n\}$, and suppose that every element in $x\in G$ can be uniquely written as $x=\prod_{i=1}^n g_i^{\epsilon_i}$. The $t$-graph of distances of $G$ is defined as the graph with vertices set $G$, and in which two vertices $x=\prod_{i=1}^n g_i^{\epsilon_i}$ and $y=\prod_{i=1}^n g_i^{\delta_i}$ are adjacent if the Minkowski distance between them is equal to $t$. That is, $l_1(x,y) = \sum_{i=1}^n |\epsilon_i-\delta_i| =t$. If $\mathscr{C}$ is subgroup of $G$, then we say that $\mathscr{C}$ is a group code. In this talk we consider such codes and the connection with $t$-graph of distances of $G$, when $t=1$.

Joint work with Elias Claro, UAM, CDMX.