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In the theory of cocyclic self-dual codes, three types of equivalences are encountered: cohomology or the equivalence of cocycles, Hadamard equivalence or the equivalence of Hadamard matrices, and the equivalence of binary linear codes. There are some results relating the latter two equivalences, see Ozeki [12], but not when the Hadamard matrices are un-normalised.
Recently, Horadam [9] discovered shift action, whereby every finite group \( G \) acts as a group of automorphisms of \( Z = Z^2(G, C) \), the finite abelian group of cocycles from \( G \times G \to C \), for each abelian group \( C \). These automorphisms fix the subgroup of coboundaries \( B \leq Z \) setwise. This shift action of \( G \) on \( Z \) partitions each cohomology class of \( Z \).
Here we show that shift-equivalent cocycles generate equivalent Hadamard matrices and that shift-equivalent cocyclic Hadamard matrices generate equivalent binary linear codes.