The weight spectrum of two families of Reed-Muller codes (Q6172304)

The Reed Muller codes are a family of binary codes of length \(2^m\), denoted by \(RM(r,m)\) where the dimension is \(\sum_{i=0}^r \binom{m}{i}\) and the minimum distance is \(2^{m-r}.\) The authors develop a technique for determining the weight spectra of these codes. It uses the fact that the sum of two weights in \(RM(r-1,m-1)\) is a weight in \(RM(r,m)\), the characterization of \textit{T. Kasami} and \textit{N. Tokura} [IEEE Trans. Inf. Theory 16, 752--759 (1970; Zbl 0205.20604)] of the weights in \(RM(r,m)\) whose weights lie between the minimum distance and double the minimum distance, and induction on \(m\). With this technique, they determine the weight spectra of \(RM(m-3,m)\) for \(m\geq 6\), \(RM(m-4,m)\) for \(m\geq8\), \(RM(3,8)\), and \(RM(4,9)\). They concluded with a conjecture on the weights of \(RM(m-c,m)\) where \(c\) is a constant and \(m\) is sufficiently large.