Second weight codewords of generalized Reed-Muller codes
Cryptography and Communications
| Hi-index | 754.84 |
The following theorem is proved. Letf(x_1,cdots, x_m)be a binary nonzero polynomial ofmvariables of degreenu. 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+muandnu 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+2mu -3} y_{nu+2mu-2}), end{equation} This theorem completely characterizes the codewords of thenuth-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.