Nonstandard linear recurring sequence subgroups in finite fields and automorphisms of cyclic codes

Let q = pr be a prime power, and let f(x) = xm − σm−1x m−1 − · · · − σ1x− σ0 be an irreducible polynomial over the finite field GF(q) of size q. A zero ξ of f is called nonstandard (of degree m) over GF(q) if the recurrence relation um = σm−1um−1 + · · · + σ1u1 + σ0u0 with characteristic polynomial f can generate the powers of ξ in 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 irreducible polynomials f of the form f(x) = xm − f0, and is well-understood. The other class (type II) can be obtained from a primitive element in some subfield by a process that we call extension and lifting. We will use the known classification of the subgroups of PGL(2, q) in combination with a recent result by Brison and Nogueira to show that a nonstandard element of degree two over GF(q) necessarily is of type I or type II, thus solving completely the classification problem for the case m = 2.