Zentralblatt für Mathematik
Mathematics Abstracts

857.11065

Kaplan, Michael; Staiger, Ludwig:
Low-degree Mattson-Solomon polynomials.
[J] Atti Semin. Mat. Fis. Univ. Modena 43, No.2, 265-272 (1995). [ISSN 0041-8986]

Classification
*11T06 Polynomials over finite fields or rings
11T71 Algebraic coding theory
94B15 Cyclic codes
Keywords:
cyclic codes; Mattson-Solomon polynomial; finite fields

Let $GF(q)$ be the finite field with $q$ elements, $n$ a positive integer relatively prime to $q$ and $\alpha$ a primitive $n$-th root of unity in some extension field of $GF(q)$. If $c(x) = \sum\sp{n-1}\sb{i=0} c\sb ix\sp i \in GF(q) [x]$ then its Mattson-Solomon (MS) polynomial is $g(x) = \sum\sp{n-1}\sb{j=0} c(\alpha\sp{-j}) z\sp j$. The following theorem is established:

Theorem. Let $g(z)$ be a polynomial of degree $\le \min \{n-1, 2n/q\}$ with coefficients in some extension field of $GF(q)$. Then $g(z)$ is an MS polynomial of some polynomial $c(x) \in GF(q) [x]$ of degree $\le n-1$ iff $$g(z)\sp q-g(z) = -{z \over n} (z\sp n-1) g'(z).$$

This theorem extends the binary version due to {\it A. Kerdock}, {\it F. J. MacWilliams} and {\it A. Odlyzko} [IEEE Trans. Inf. Theory 20, 85-89 (1974; Zbl 277.94007)]. A corollary to the theorem states that if $g(z)$ is a polynomial of degree $\le \min \{n-1,2n/q\}$ and simultaneously an MS polynomial of some polynomial $c(x) \in GF(q)[x]$ then $\deg g(z) \le n/(q-1)$.
I.F. Blake (Palo Alto)

Publ. Year: 1995
Document Type: Journal

(c) 1996 FIZ Karlsruhe & Springer-Verlag