On the second weight of generalized Reed-Muller codes

Authors:
Olav Geil
Affiliations:
Department of Mathematical Sciences, Aalborg University, Aalborg, Denmark
Venue:
Designs, Codes and Cryptography
Year:
2008

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Abstract

Not much is known about the weight distribution of the generalized Reed-Muller code RM q (s,m) when q 2, s 2 and m 驴 2. Even the second weight is only known for values of s being smaller than or equal to q/2. In this paper we establish the second weight for values of s being smaller than q. For s greater than (m --- 1)(q --- 1) we then find the first s + 1 --- (m --- 1)(q---1) weights. For the case m = 2 the second weight is now known for all values of s. The results are derived mainly by using Gröbner basis theoretical methods.