Let q = pr be a prime power, and let f(x) = xm − �m−1xm−1 − � � � − �1x − �0 be an irreducible polynomial over the finite field GF(q) of size q. A zeroof f is called nonstandard (of degree m) over GF(q) if the recurrence relation um =...
moreLet q = pr be a prime power, and let f(x) = xm − �m−1xm−1 − � � � − �1x − �0 be an irreducible polynomial over the finite field GF(q) of size q. A zeroof f is called nonstandard (of degree m) over GF(q) if the recurrence relation um = �m−1um−1 + � � � + �1u1 + �0u0 with characteristic polynomial f can generate the powers ofin a nontrivial way, that is, with u0 = 1 and f(u1) 6= 0. In 2003, Brison and Nogueira asked for a characterisation of all nonstandard cases in the case m = 2, and solved this problem for q a prime, and later for q = pr with r ≤ 4. In this paper, we first show that classifying nonstandard finite field elements is equivalent to classifying those cyclic codes over GF(q) generated by a single zero that posses extra permutation automorphisms. Apart from two sporadic examples of degree 11 over GF(2) and of degree 5 over GF(3), related to the Golay codes, there exist two classes of examples of nonstandard finite field elements. One of these classes (type I) involves irr...
