مرکز منطقه ای اطلاع رساني علوم و فناوري - A new theorem about the Mattson-Solomon polynomial, and some applications

Abstract :

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.