On the Weight Distributions of Optimal Cosets
of the First-Order
Reed-Muller Codes

Anne Canteaut
INRIA, projet CODES
BP 105
78153 Le Chesnay Cedex, France Anne.Canteaut@inria.fr

IEEE Transactions on Information Theory, 47(1):407-413, January 2001.

Abstract

We study the weight distributions of cosets of the first-order
Reed-Muller code R(1,m) for odd m, whose minimum weight is
greater than or equal to the so-called quadratic bound. Some general
restrictions on the weight distribution of a coset of R(1,m) are
obtained by partitioning its words according to their weight
divisibility. Most notably, we show that there are exactly five
weight distributions for optimal cosets of R(1,7) in R(5,7)
and that these distributions are related to the degree of the
function generating the coset. Moreover, for any odd m >= 9, we
exhibit optimal cubic cosets of R(1,m) whose weights take on
exactly five values.