We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of projective Reed-Muller codes for some particular degrees that give codes with good parameters. Moreover, from this recursive construction we derive a lower bound for the generalized Hamming weights of projective Reed-Muller codes which is sharp in most of the cases we have checked.

翻译：我们给出射影Reed-Muller码的一种递归构造，该构造通过仿射Reed-Muller码和变量数更少的射影Reed-Muller码实现。基于此构造，我们得到了在某些特定次数下射影Reed-Muller码的子域子码的维数，这些参数能产生具有良好性能的编码。此外，从该递归构造出发，我们推导出射影Reed-Muller码的广义汉明权重的下界，该下界在我们验证的大多数情况下是紧的。