We give a complete characterization of all Galois subfields of the generalized GiuliettiKorchmáros function fields Cn/F q 2n for n ≥ 5. Calculating the genera of the corresponding fixed fields, we find new additions to the list of known genera of maximal function fields.

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For an odd prime p and an even integer n with gcd(n, p) = 1, we consider quadratic functions from F p n to F p of codimension k. For various values of k, we obtain classes of quadratic functions giving rise to maximal and minimal Artin-Schreier curves over F p n . We completely classify all maximal and minimal curves obtained from quadratic functions of codimension 2 and coefficients in the prime field F p .

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Bicovering arcs in Galois affine planes of odd order are a powerful tool for the construction of complete caps in spaces of arbitrarily higher dimensions. The aim of this paper is to investigate whether the arcs contained in elliptic cubic curves are bicovering. As a result, bicovering k-arcs in AG(2, q) of size k ≤ q/3 are obtained, provided that q − 1 has a prime divisor m with 7 < m < (1/8)q 1/4 . Such arcs produce complete caps of size kq (N −2)/2 in affine spaces of dimension N ≡ 0 (mod 4). When q = p h with p prime and h ≤ 8, these caps are the smallest known complete caps in AG(N, q), N ≡ 0 (mod 4).