On weight spaces of polynomial representations of the general linear group as linear codes

Abstract

We provide a basis for the weight spaces of certain polynomial representations of the general linear group introduced by G. James. Then we determine the minimum distance of those weight spaces which have highest error correction capabilities among all the studied weight spaces and derive a new class of completely majority-logic decodable linear codes. Finally, we show that binary Reed-Muller and Simplex codes also occur as weight spaces.