On the weight structure of Reed-Muller codes

TLDR: This theorem completely characterizes the codewords of the \nu th-order Reed-Muller code whose weights are less than twice the minimum weight and leads to the weight enumerators for thosecodewords.

Abstract: The following theorem is proved. Let f(x_1,\cdots, x_m) be a binary nonzero polynomial of m variables of degree \nu . H the number of binary m -tuples (a_1,\cdots, a_m) with f(a_1, \cdots, a_m) = 1 is less than 2^{m-\nu+1} , then f can be reduced by an invertible affme transformation of its variables to one of the following forms. \begin{equation} f = y_1 \cdots y_{\nu - \mu} (y_{\nu-\mu+1} \cdots y_{\nu} + y_{\nu+1} \cdots y_{\nu+\mu}), \end{equation} where m \geq \nu+\mu and \nu \geq \mu \geq 3 . \begin{equation} f = y_1 \cdots y_{\nu-2}(y_{\nu-1} y_{\nu} + y_{\nu+1} y_{\nu+2} + \cdots + y_{\nu+2\mu -3} y_{\nu+2\mu-2}), \end{equation} This theorem completely characterizes the codewords of the \nu 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.