On the third weight of generalized Reed-Muller codes

In this paper, we study the third weight of generalized Reed-Muller codes. We prove under some restrictive condition that the third weight of generalized Reed-Muller codes depends on the third weight of generalized Reed-Muller codes of small order with two variables. In some cases, we are able to determine the third weight and the third weight codewords of generalized Reed-Muller codes.