A characterization of abelian group codes in terms of their parameters

In 1979, Miller proved that for a group $G$ of odd order, two minimal codes in $\mathbb{F}_2G$ are $G$-equivalent if and only they have identical weight distribution. 2014, Ferraz-Guerreiro-Polcino Milies disprove Miller's result by giving an example non-$G$-equivalent with this paper, we give characterization finite abelian groups so over specific set codes, equality important parameters implies the $G$-equivalence these codes. As corollary, prove same distribution each prime divisor $p$ $|G|$, Sylow $p$-subgroup is homocyclic.