An Exploration of Absolute Minimal Degree Lifts of Hyperelliptic Curves

For any ordinary elliptic curve E over a field with non-zero characteristic p, there exists an elliptic curve E over the ring of Witt vectors W(E) for which we can lift the Frobenius morphism, called the canonical lift. Voloch and Walker used this theory of canonical liftings of elliptic curves over Witt vectors of length 2 to construct non-linear error-correcting codes for characteristic two. Finotti later proved that for longer lengths of Witt vectors there are better lifts than the canonical. He then proved that, more generally, for hyperelliptic curves one can construct a lifting over the ring of Witt vectors with minimal degrees of the coordinate polynomials of the lift of points, and that these achieve better parameters for the resulting non-linear codes. In this dissertation we describe an algorithm to compute these absolute minimal degree liftings of hyperelliptic curves, along with constructing non-linear codes over finite fields of non-zero characteristic greater than two. From our algorithm and exploratory analysis we derive results and conjectures regarding requirements to compute absolute minimal degree lifts satisfying the theoretical lower bound, i.e., the ones that give the best bounds for the parameters of the resulting code, and compare our resulting non-linear codes with currently best known linear error-correcting codes.