Cyclic codes and the Frobenius automorphism
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 \(m\).
scientific article; zbMATH DE number 1292446