Summary: In this paper, by introducing the concept of consulting designs and based on the connection between factorial design theory and coding theory, the authors obtain some combinatorial identities that relate the word length pattern of a regular mixed-level \((2^\gamma)2^n\) factorial to that of its consulting design. Consequently, a general rule for identifying minimum aberration \((z^\gamma)z^n\) factorial designs through their consulting designs is established. This is an improvement and generalization of the related results of R. Mukerjee and C.F.J. Wu [Stat. Sin. 11, No. 1, 225–239 (2001; Zbl 0967.62054)]