Abstract Let F be any field containing the finite field of order q . A q -polynomial L over F is an element of the polynomial ring F [ x ] with the property that all powers of x that appear in L with nonzero coefficient have exponent a power of q . It is well known that given any ordinary polynomial f in F [ x ], there exists a q -polynomial that is divisible by f . We study the smallest degree of such a q -polynomial. This is equivalent to studying the $${{\,\mathrm{\mathbb {F}}\,}}_q$$ F q -span of the roots of f in a splitting field. We relate this quantity to the representation theory of the Galois group of f . As an application we give a simultaneous construction of the binary Golay code of length 24, and the Steiner system on 24 points.

The triple autocorrelation of a sequence is shown to be a Lempel Costas array iff the sequence is maximal, assuming correlation is appropriately defined and the alphabet of the sequence is of prime size. A method that allows the construction of thumbtack autocorrelation sequences of arbitrary alphabet size is subsequently proposed.

Families of Costas arrays with low pairwise cross-correlation are sought. The two families of all exponential Welch arrays and all Golomb arrays generated in a certain finite field are specifically studied, and the maximal cross-correlation is determined by exhaustive search. Mathematically rigorous explanations for some of the observed results are presented, a surprising link between Welch and Golomb arrays is revealed, and what remains to be proved is stated precisely. The results suggest that the families with uniformly low cross-correlation correspond to finite fields whose size is a safe prime power.