On the relation of character codegrees and the minimal faithful quasi-permutation representation degree of $p$-groups

For a finite group $G$, we denote by $c(G)$, the minimal degree of faithful representation of $G$ by quasi-permutation matrices over the complex field $\mathbb{C}$. For an irreducible character $\chi$ of $G$, the codegree of $\chi$ is defined as $\text{cod}(\chi )=\mathrm{\mid }G\mathrm{/}\ker (\chi )\mathrm{\mid }\mathrm{/}\chi (1)$. In this article, we establish equality between $c(G)$ and a ${\mathbb{Q}}_{\ge 0}$-sum of codegrees of some irreducible characters of a non-abelian $p$-group $G$ of odd order. We also study the relation between $c(G)$ and irreducible character codegrees for various classes of non-abelian $p$-groups, such as, $p$-groups with cyclic center, maximal class $p$-groups, GVZ $p$-groups, and others.