On the third weight of generalized Reed-Muller codes

In this paper, we study the third weight of generalized Reed-Muller codes. Using results from Erickson, we prove under some 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. We are able to determine the third weight and the third weight codewords of generalized Reed-Muller codes of order r=a(q−1)+b with 2≤b≤q−1, for small b.