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Let \(C\) be a perfect 1-error-correcting code of length \(15\). We show that a quotient \(H(C)\) of the minimum distance graph of \(C\) constitutes an invariant for \(C\) more sensible than those studied up to the present, namely the kernel dimension and the rank. As a by-product, we get a nonlinear Vasil’ev code \(C\) all of whose associated Steiner triple systems are linear. Finally, the determination of \(H(C)\) for known families of \(C\)’s is presented.