Further results on the semilinear equivalence of linear codes

Based on relative subcodes, we address an equivalent condition for a one-to-one semilinear mapping between two linear codes to be in fact a semimonomial transformation, that is, the underlying two codes are semilinearly equivalent. The result in the present paper substantially improves the equivalent condition in recent literatures. Moreover, it also generalizes the well-known MacWilliams theorem of code equivalence.