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Session S29 - Theory and Applications of Coding Theory

Monday, July 19, 18:30 ~ 18:55 UTC-3

Weierstrass pure gaps and codes on curves with three distinguished points

Herivelto Borges

Universidade de Sao Paulo, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak80e3816b6fd83219bf4264c75cf593aa').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy80e3816b6fd83219bf4264c75cf593aa = 'hb&#111;rg&#101;s' + '&#64;'; addy80e3816b6fd83219bf4264c75cf593aa = addy80e3816b6fd83219bf4264c75cf593aa + '&#105;cmc' + '&#46;' + '&#117;sp' + '&#46;' + 'br'; var addy_text80e3816b6fd83219bf4264c75cf593aa = 'hb&#111;rg&#101;s' + '&#64;' + '&#105;cmc' + '&#46;' + '&#117;sp' + '&#46;' + 'br';document.getElementById('cloak80e3816b6fd83219bf4264c75cf593aa').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy80e3816b6fd83219bf4264c75cf593aa + '\'>'+addy_text80e3816b6fd83219bf4264c75cf593aa+'<\/a>';

In this talk we consider the class of smooth plane curves of degree $n+1>3$ over a finite field containing three points, $P_1,P_2,$ and $P_3$, such that $nP_1+P_2$, $nP_2+P_3$, and $nP_3+P_1$ are divisors cut out by three distinct lines. For any such a curve, the dimension of certain special divisors supported on $\{P_1,P_2,P_3\}$ is computed, and an explicit description of the set of all pure gaps at any subset of $\{P_1,P_2,P_3\}$ is provided. From this class of curves, which includes the Hermitian curve, one can construct Goppa codes having minimum distance better than the Goppa bound.

Joint work with Gregory Cunha (Universidade Federal de Goias).