## View abstract

### 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. document.getElementById('cloaka017abf447b99dec34aa2c7bb51bbf4e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addya017abf447b99dec34aa2c7bb51bbf4e = '&#105;sg&#117;t&#105;&#101;r' + '&#64;'; addya017abf447b99dec34aa2c7bb51bbf4e = addya017abf447b99dec34aa2c7bb51bbf4e + '&#117;n&#105;n&#111;rt&#101;' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;'; var addy_texta017abf447b99dec34aa2c7bb51bbf4e = '&#105;sg&#117;t&#105;&#101;r' + '&#64;' + '&#117;n&#105;n&#111;rt&#101;' + '&#46;' + '&#101;d&#117;' + '&#46;' + 'c&#111;';document.getElementById('cloaka017abf447b99dec34aa2c7bb51bbf4e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addya017abf447b99dec34aa2c7bb51bbf4e + '\'>'+addy_texta017abf447b99dec34aa2c7bb51bbf4e+'<\/a>';

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.