Isomorphism classes of hyperelliptic curves of genus 3 over finite fields

In this paper we present a direct method to compute the number of isomorphism classes of hyperelliptic curves of genus 3 over finite fields . The number of isomorphism classes is a polynomial in q of degree 5. In all the cases we show an explicit formula for this polynomial. These results can be used in the classification problems and the hyperelliptic curve cryptosystems.