LetF = GF(2), andFG = F[x]/(x^n + 1). FGis the residue class ring of polynomials modx^n + 1. An element ofFGis represented by a polynomial of degree at mostn - 1\begin{equation} c(x) = c_0 + c_1 x + \cdots + c_{n-1} x^{n-1} \end{equation} with coefficients inF. 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 inGF(2^m), wheremis the least integer such thatndivides2^m - 1, and\alphais a primitiventh 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. Letn = 2^m - 1, wherem \equiv 1mod 2. Let\mathcal{A}_1be the cyclic code of dimension 2m defined by the property that its check polynomial has zeros\alpha ^{-j}, wherej = 1,2,\cdots,2^{m-1}andj = l,2l,\cdots,2^{m-1} l, l = 2^i + 1. If(i,m) = 1this code has just three nonzero weights, namely,2^{m-1} \pm 2^{(m-1)/2}and2^{m-1}. The weight distribution can then be obtained from the MacWflliams identifies. These conditions are satisfied forn = 31, l = 3,5; n = 127,l= 3,5,9;n = 511, l = 3,5,17; etc. Thus forn= 127, for example, the three codes\mathcal{A}_3,\mathcal{A}_5, \mathcal{A}_9have the same weight distribution, although they are probably not equivalent in the usual sense.

It is shown that the mixture of two codes of block lengthncan have a distortion strictly less than the distortion of the best possible code of block lengthnhaving the same rate. The seeming contradiction is explained by offering two interpretations for the mixture of two codes and then explaining why the bound for the performance of the best code cannot be applied directly to the mixture.