Key points are not available for this paper at this time.
The following theorem is proved. Let f (x₁, , xₘ) be a binary nonzero polynomial of m variables of degree. H the number of binary m -tuples (a₁, , aₘ) with f (a₁, , aₘ) = 1 is less than 2^m-+1, then f can be reduced by an invertible affme transformation of its variables to one of the following forms. equation f = y₁ y - (y-+₁ y_ + y+₁ y+), equation where m + and 3. equation f = y₁ y-₂ (y-₁ y_ + y+₁ y+₂ + + y+₂ -₃ y+₂-₂), equation This theorem completely characterizes the codewords of the th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for those codewords. These weight formulas are extensions of Berlekamp and Sloane's results.
Kasami et al. (Sun,) studied this question.