Cyclic codes and the Frobenius automorphism
type: Item
instance of: scholarly article
title: Cyclic codes and the Frobenius automorphism
zbMATH Open document ID: 0928.94012
author: Arne Winterhof
published in: Archiv der Mathematik
publication date: 1999-08-22T00:00:00Z
review text: The main theorem of this paper provides the complete (and perhaps unexpected) solution of a problem posed by \textit{K. Burde} [J. Reine Angew. Math. 278/279, 353-364 (1975; Zbl 0323.12019)]. Let \(q=p^f\) be a prime power, \(n=p^bm\) with \(gcd(p,m)=1\). Consider \(K=GF(q^n)\) is an \(n\)-dimensional vector space over \(F=GF(q)\). Let \(\sigma\) generate the Galois group \({\mathcal G}(K/F)\). The main result says that for each \(k\), \(0\leq k\leq n\), there is a unique vector subspace \(C\) of \(K\) having dimension \(k\) and which is invariant under \(\sigma\) if and only if \(m=1\) or \(m=p^b+2\) is prime and \(q\) is a primitive root modulo \(m\).
Mathematics Subject Classification ID: 94B15
Mathematics Subject Classification ID: 11T30
zbMATH DE Number: 1292446
zbMATH Keywords: cyclic code
zbMATH Keywords: finite field
zbMATH Keywords: Galois group
zbMATH Keywords: vector subspace
zbMATH Keywords: primitive root
reviewed by: Stanley E. Payne
MaRDI profile type: Publication
full work available at URL: s000130050301
OpenAlex ID: W2056038930
DOI: 10.1007/S000130050301
description: scientific article; zbMATH DE number 1292446