On the Hermitian curve, its intersections with some conics and their applications to affine-variety codes and Hermitian codes

Abstract

For any affine-variety code we show how to construct an ideal whose solutions correspond to codewords with any assigned weight. We classify completely the intersections of the Hermitian curve with lines and parabolas (in the ${\mathbb{F}}_{q^2}$ affine plane). Starting from both results, we are able to obtain geometric characterizations for small-weight codewords for some families of Hermitian codes over any ${\mathbb{F}}_{q^2}$. From the geometric characterization, we obtain explicit formulae. In particular, we determine the number of minimum-weight codewords for all Hermitian codes with $d\le q$ and all second-weight codewords for distance-$3,4$ codes.

References in corpus (2)

• The dual minimum distance of arbitrary dimensional algebraic--geometric codes
• Improved decoding of affine-variety codes

Cited by in corpus (1)

• On the Hermitian curve and its intersections with some conics