Global construction of general exceptional covers, with motivation for applications to coding Download/info .dvi or .pdf: click here.

Last rev: 10/23/2000: G.L.~Mullen and P.J.~Shiue edit., Finite Fields: Theory, applications and algorithms, Cont.~Math. 168 (1994), 69--100. A cover f:X --> P¹ (of projective non-singular curves over a finite field F_q) is general exceptional if f is one-one on F_q^t points of X for infinitely many t. Exceptional polynomials have been around for 140 years. We generalize, applying the method of Fried-Völklein to go from the equivalent monodromy group conditions to producing general exceptional covers over all finite fields in great abundance (and of arbitrary genus). Problem: General exceptional covers include median value curves: Having exactly q^t+1 points over F_q^t points for infinitely many t. This condition holds for t a union of arithmetic progressions. It poses geometrically relating a median value curve to some exceptional cover.