Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 5, No 1 (2011)

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Genus calculations for towers of function fields arising from equations of $C_{ab}$ curves

Caleb McKinley Shor


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.

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