On the dual code of points and generators on the Hermitian variety $\mathcal{H}(2n+1,q^2)$

TL;DR

This paper investigates the dual linear code associated with points and generators on Hermitian varieties, providing new classifications, solving the minimum distance problem for all dimensions, and characterizing small weight code words.

Contribution

It advances understanding of the dual code structure on Hermitian varieties by solving the minimum distance problem for general n and classifying small weight code words.

Findings

Improved results for the case n=2.

Solved the minimum distance problem for all n.

Characterized small weight code words as linear combinations of n types.

Abstract

We study the dual linear code of points and generators on a non-singular Hermitian variety $\mathcal{H}(2n+1,q^2)$. We improve the earlier results for $n=2$, we solve the minimum distance problem for general $n$, we classify the $n$ smallest types of code words and we characterize the small weight code words as being a linear combination of these $n$ types.