Explicit bases for the subfield subcodes of projective Reed-Muller codes over the projective plane and their duals are obtained. In particular, we provide a formula for the dimension of these codes. For the general case over the projective space, we generalize the necessary tools to deal with this case as well: we obtain a universal Gr\"obner basis for the vanishing ideal of the set of standard representatives of the projective space and we show how to reduce any monomial with respect to this Gr\"obner basis. With respect to the parameters of these codes, by considering subfield subcodes of projective Reed-Muller codes we obtain long linear codes with good parameters over a small finite field.

翻译：本文得到了射影平面上射影Reed-Muller码的子域子码及其对偶码的显式基，并特别给出了这些码的维数公式。对于射影空间中的一般情形，我们推广了处理该情形所需的基本工具：得到了射影空间标准代表集消失理想的通用Gröbner基，并展示了如何利用该Gröbner基化简任意单项式。关于这些码的参数，通过考虑射影Reed-Muller码的子域子码，我们在小有限域上获得了具有良好参数的长线性码。