A new theorem about the Mattson-Solomon polynomial, and some applications

Let F = GF(2) , and FG = F[x]/(x^n + 1). FG is the residue class ring of polynomials mod x^n + 1 . An element of FG is represented by a polynomial of degree at most n - 1 \begin{equation} c(x) = c_0 + c_1 x + \cdots + c_{n-1} x^{n-1} \end{equation} with coefficients in F . It may also be represented by a polynomial \begin{equation} g(z) = \sum_{j=0}^{n-1} c(\alpha^j)z^j \end{equation} with coefficients in GF(2^m) , where m is the least integer such that n divides 2^m - 1 , and \alpha is a primitive n th root of unity. Mattson and Solomon [1] introduced this representation in 1961. The new theorem states that \begin{equation} zg'(z) = \frac{g(z)(g(z) + 1)}{z^n +l}. \end{equation} A typical application of this result is as follows. Let n = 2^m - 1 , where m \equiv 1 mod 2. Let \mathcal{A}_1 be the cyclic code of dimension 2m defined by the property that its check polynomial has zeros \alpha ^{-j} , where j = 1,2,\cdots , 2^{m-1} and j = l,2l,\cdots,2^{m-1} l, l = 2^i + 1 . If (i,m) = 1 this code has just three nonzero weights, namely, 2^{m-1} \pm 2^{(m-1)/2} and 2^{m-1} . The weight distribution can then be obtained from the MacWflliams identifies. These conditions are satisfied for n = 31, l = 3,5; n = 127 , l = 3,5,9; n = 511, l = 3,5,17 ; etc. Thus for n = 127, for example, the three codes \mathcal{A}_3,\mathcal{A}_5, \mathcal{A}_9 have the same weight distribution, although they are probably not equivalent in the usual sense.