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 are able to derive a lower bound for the generalized Hamming weights of projective Reed-Muller codes which, together with the basic properties of the generalized Hamming weights, allows us to determine most of the weight hierarchy of projective Reed-Muller codes in many cases.

翻译：本文给出了一种关于射影Reed-Muller码的递归构造，该构造基于仿射Reed-Muller码与较少变量下的射影Reed-Muller码。通过此构造，我们获得了某些特定次数下射影Reed-Muller码子域子码的维数，这些子码具有优良的参数。此外，利用该递归构造，我们推导出了射影Reed-Muller码广义汉明权重的下界，该下界结合广义汉明权重的基本性质，使得在多数情况下能够确定射影Reed-Muller码的大部分重量层级。