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 are able to 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 are able 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 are able to obtain long linear codes with good parameters over a small finite field.

翻译：本文得到了射影平面上投影Reed-Muller码的子域子码及其对偶码的显式基。特别地，我们给出了这些码维数的计算公式。针对射影空间的一般情形，我们推广了处理该情形所需的必要工具：得到了射影空间标准代表集消去理想的通用Gröbner基，并能够将该Gröbner基下任意单项式化为标准形。关于这些码的参数，通过考虑投影Reed-Muller码的子域子码，我们能够在小有限域上构造出具有良好参数的长线性码。