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 $\mathrm{\Delta }$. 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 $\le (4-1\mathrm{/}(2n)){\log }^2\mathrm{\Delta }$, except for a finite number of fields of degree $n\le 4$. For those fields, the conclusion holds with the slightly larger limit $(4-1\mathrm{/}(2n)+1\mathrm{/}(2n^2)){\log }^2\mathrm{\Delta }$. When the cardinality of $\mathcal{C}\text{\hspace{0.17em}⁣}\mathrm{\ell }$ is odd the bounds improve to $(4-2\mathrm{/}(3n)){\log }^2\mathrm{\Delta }$, again with finitely many exceptions in degree $n\le 4$, and to $(4-2\mathrm{/}(3n)+3\mathrm{/}(8n^2)){\log }^2\mathrm{\Delta }$ without exceptions.