On the weight structure of Reed-Muller codes

The following theorem is proved. Letf(x_1,\cdots, x_m)be a binary nonzero polynomial ofmvariables of degree\nu. H the number of binarym-tuples(a_1,\cdots, a_m)withf(a_1, \cdots, a_m)= 1 is less than2^{m-\nu+1}, thenfcan 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} wherem \geq \nu+\muand\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\nuth-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.