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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('cloak5cc01b7e71d4722f8d20db8e4cdf213f').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy5cc01b7e71d4722f8d20db8e4cdf213f = 'y&#117;r&#105;k&#111;.p&#105;t&#111;n&#101;s' + '&#64;'; addy5cc01b7e71d4722f8d20db8e4cdf213f = addy5cc01b7e71d4722f8d20db8e4cdf213f + 'c&#105;m&#97;t' + '&#46;' + 'mx'; var addy_text5cc01b7e71d4722f8d20db8e4cdf213f = 'y&#117;r&#105;k&#111;.p&#105;t&#111;n&#101;s' + '&#64;' + 'c&#105;m&#97;t' + '&#46;' + 'mx';document.getElementById('cloak5cc01b7e71d4722f8d20db8e4cdf213f').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy5cc01b7e71d4722f8d20db8e4cdf213f + '\'>'+addy_text5cc01b7e71d4722f8d20db8e4cdf213f+'<\/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.