Characters and the equivalence of codes (Q1906145)

Let \(K\) be a finite field. A linear code of dimension \(k\) and block length \(n\) is a \(k\)-dimensional \(K\)-linear subspace \(C\) of the \(n\)-dimensional vector space \(K^n\) over \(K\). Two linear codes \(C_1\) and \(C_2\) in \(K^n\) are equivalent if there exists a monomial transformation \(T: K^n\to K^n\) such that \(T(C_1)= C_2\). MacWilliams proved that two linear codes are equivalent if and only if there is a weight-preserving linear isomorphism between them. In this paper the authors present a new proof based on the linear independence of group characters. As an application, the authors also give another proof of a theorem of Bonisoli characterizing constant weight linear codes.