GL(m, 2) acting on R(r, m)/R(r − 1, m)

✍ Scribed by Xiang-dong Hou

• Book ID: 103060903
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 895 KB
• Volume: 149
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(94)00342-g

✦ Synopsis

Let R(r,m) be the rth order Reed-Muller code of length 2", and let R(r,m)/R(r -1,m) be the set of all eosets of R(r-1,m) in R(r,m). The general linear group GL(m,2) acts on R(r,m)/R(r-1,m). We compute the numbers of the GL(m,2)-orbits of R(r,m)/R(r-1,m) for 6 ~< m ~< 11. This is done through a formula for the size of the centralizer of a matrix in GL(m, 2) and the observation that any A E GL(m,2) acts on R(r,m)/R(rl,m) as a linear isomorphism whose matrix with respect to the basis {l--LEsXi +R(r-1,m) :SC{1 ..... m},[S I = r} of R(r, m)/R(r-1, m) is the rth compound of A. We then classify R(3, 6)/R(2, 6) and R(3, 7)/R(2, 7 ).

The implication of these classifications concerning the covering properties of R(2,6) and R(2, 7) is also given.