Quasi-quadratic residue codes and hyperelliptic curves
Nigel Boston, Jing Hao · Contemporary mathematics - American Mathematical Society · 2019
We introduce a family of codes called quasi-quadratic residue (QQR) codes in algebraic coding theory. We are interested in these codes because their weight distributions play an important role in a famous conjecture in coding theory known as Goppa’s conjecture . We summarize some existing results on their weight distributions. Furthermore, the weight of their codewords have a close relation with the number of points on corresponding hyperelliptic curves. In the second part, we implement a heuristic model to estimate the limiting behaviour of number of points on related hyperelliptic curves over F p \mathbb {F}_p as p p goes to infinity. For primes p > 50 p>50 , explicit calculations are given. These results provide evidence that Goppa’s conjecture is dubious.