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 . [ABSTRACT FROM AUTHOR]