Breaking the 4 barrier for the bound of a generating set of the class group

Let K be a field of degree n and discriminant with absolute value Delta. Under the assumption of the validity of the Generalized Riemann Hypothesis, we provide a new algorithm to compute a set of generators of the class group of K and prove that the norm of the ideals in that set is <= (4 - 1/(2n)) log(2) Delta, except for a finite number of fields of degree n <= 4. For those fields, the conclusion holds with the slightly larger limit (4-1/(2n)+1/(2n(2))) log(2) Delta. When the cardinality of Cl is odd the bounds improve to (4 - 2/(3n)) log(2) Delta, again with finitely many exceptions in degree n <= 4, and to (4 - 2/(3n) + 3/(8n(2))) log(2) Delta without exceptions.