## View abstract

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

Wednesday, July 14, 13:00 ~ 13:25 UTC-3

## Algebraic interpretation of the minimum distance of Reed-Muller-type codes

### Yuriko Pitones

#### CIMAT , Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak364448902cf9ae75bb1c43ab64f9e907').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy364448902cf9ae75bb1c43ab64f9e907 = 'y&#117;r&#105;k&#111;.p&#105;t&#111;n&#101;s' + '&#64;'; addy364448902cf9ae75bb1c43ab64f9e907 = addy364448902cf9ae75bb1c43ab64f9e907 + 'c&#105;m&#97;t' + '&#46;' + 'mx'; var addy_text364448902cf9ae75bb1c43ab64f9e907 = 'y&#117;r&#105;k&#111;.p&#105;t&#111;n&#101;s' + '&#64;' + 'c&#105;m&#97;t' + '&#46;' + 'mx';document.getElementById('cloak364448902cf9ae75bb1c43ab64f9e907').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy364448902cf9ae75bb1c43ab64f9e907 + '\'>'+addy_text364448902cf9ae75bb1c43ab64f9e907+'<\/a>';

The minimum distance of Reed-Muller-type codes has an algebraic interpretation, in terms of its associated vanishing ideal $I$ and the Hilbert-Samuel multiplicity of $I$, called the $\delta$-function. In this talk, we present this interpretation and study the asymptotic behavior of the $\delta$-function, in particular, we related the stabilization point, $r$ of $I$ of the $\delta$-function with the Castelnuovo-Mumford regularity of $I$. We see that when generalized the $\delta$-function to graded ideals, the point $r I$ is less than or equal to the Castelnuovo -Mumford regularity of $I$, in particular, this claim holds for F-pure and square free monomial ideals.