IJPAM: Volume 97, No. 2 (2014)

GENERALIZED HAMMING WEIGHTS OF
DUALS OF ALGEBRAIC-GEOMETRIC CODES

E. Ballico
Department of Mathematics
University of Trento
38 123 Povo (Trento) - Via Sommarive, 14, ITALY


Abstract. Here we extends a work of A. Couvreur on the Hamming distance of the dual of an evaluation code to its generalized Hamming weights. We prove the following result.

Fix integers $r\ge 2$, $m>0$ and $e\ge 1$. Let $Z\subset \mathbb {P}^r$ be a zero-dimensional scheme such that $\deg (Z) \le 3m+r-3$. If $r >2$ assume that $Z$ spans $\mathbb {P}^r$ and that the sum of the degrees of the non-reduced connected components of $Z$ is at most $2m+1$. We have $h^1(\mathcal {I}_Z(m)) \ge e$ if and only if there is $W\subseteq Z$ as one of the schemes in the following list:

(a)
$\deg (W) = m+1+e$ and $W$ is contained in a line;
(b)
$\deg (W) = 2m+1+e$ and $W$ is contained in a reduced plane conic;
(c)
$r\ge 3$, $e\ge 2$, and there are an integer $f\in \{1,\dots ,e-1\}$ and lines $L_1, L_2$, such that $L_1\cap L_2=\emptyset$, $\deg (L_1\cap Z) = m+1+f$ and $\deg (L_2\cap Z)=m+1+e-f$.


Received: August 6, 2014

AMS Subject Classification: 4N05, 14Q05, 94B27

Key Words and Phrases: dual code, generalized Hamming weights, higher support weights, evaluation code, plane curve, algebraic-geometric code

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DOI: 10.12732/ijpam.v97i2.13 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2014
Volume: 97
Issue: 2
Pages: 241 - 251


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