The Weight Distribution of Quasi-quadratic Residue Codes

We investigate a family of codes called quasi-quadratic residue (QQR) codes . We are interested in these codes mainly for two reasons: Firstly, they have close relations with hyperelliptic curves and Goppa's Conjecture, and serve as a strong tool in studying those objects. Secondly, they are very good codes. Computational results show they have large minimum distances when \begin{document}$p\equiv 3 \pmod 8$\end{document} . Our studies focus on the weight distributions of these codes. We will prove a new discovery about their weight polynomials, i.e. they are divisible by \begin{document}$(x^2 + y^2)^{d-1}$\end{document} , where \begin{document}$d$\end{document} is the corresponding minimum distance. We also show that the weight distributions of these codes are asymptotically normal. Based on the relation between QQR codes and hyperelliptic curves, we will also prove a result on the point distribution on hyperelliptic curves.