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

Let , and is the residue class ring of polynomials mod . An element of is represented by a polynomial of degree at most begin{equation} c(x) = c_0 + c_1 x + cdots + c_{n-1} x^{n-1} end{equation} with coefficients in . 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 , where is the least integer such that divides , and is a primitive 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 , where mod 2. Let be the cyclic code of dimension 2m defined by the property that its check polynomial has zeros , where , and . If this code has just three nonzero weights, namely, and . The weight distribution can then be obtained from the MacWflliams identifies. These conditions are satisfied for , = 3,5,9; ; etc. Thus for = 127, for example, the three codes have the same weight distribution, although they are probably not equivalent in the usual sense.