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.

中文翻译：

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超椭圆曲线上的仿射变体代码

我们估计从超椭圆曲线定义的初级单项仿射各种代码的最小距离\（{X ^ 5} + X - {Y ^ 2} \）超过\（\ mathbb {F} _7 \） 。为了估计代码的最小距离，我们应用符号计算来实现Geil和Özbudak建议的技术。对于其中一些代码，我们还获得了符号对距离。此外，获得了所构造代码的广义汉明权重的下限。所提出的用于计算广义汉明权重的方法可以应用于任何主要的单项仿射变种代码。