GENUS CALCULATIONS FOR TOWERS OF FUNCTION FIELDS ARISING FROM EQUATIONS OF Cab CURVES

Abstract

We give a generalization of error-correcting code construction from $C_{ab}$ curves by working with towers of algebraic function fields. The towers are constructed recursively, using defining equations of $C_{ab}$ curves. In order to estimate the parameters of the corresponding one-point Goppa codes, one needs to calculate the genus. Instead of using the Hurwitz genus formula, for which one needs to know about ramification behavior, we use the Riemann-Roch theorem to get an upper bound for the genus by counting the number of Weierstrass gap numbers associated to a particular divisor. We provide a family of examples of towers which meet the bound.