Classification of cosets of the Reed-Muller code R( m−3, m)

Abstract

Cosets of the Reed-Muller code R( m−3, m) are classified under the actions of GL( m,2) and GA( m,2), the latter being the automorphism group of R( m—3, m) for m⩾4. The number of cosets in each class is calculated. Orphans of R( m−3, m) are identified, and the normality of R( m−3, m) is established. A recursive formula is given for computing weight distributions of cosets of R( m−3, m). The formula gives the number of vectors of weight w in a coset of R( m−3, m) with minimal weight μ rather easily when w is not far away from μ.