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.