Affine variety codes over a hyperelliptic curve

We estimate the minimum distance of primary monomial affine variety codes defined from a hyperelliptic curve ${x^5} + x - {y^2}$ over $\mathbb{F}_7$. To estimate the minimum distance of the codes, we apply symbolic computations implementing the techniques suggested by Geil and Özbudak. For some of these codes, we also obtain the symbol-pair distance. Furthermore, lower bounds on the generalized Hamming weights of the constructed codes are obtained. The proposed method to calculate the generalized Hamming weights can be applied to any primary monomial affine variety codes.