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6. Let m be an even integer and consider the representation ρ(mZII ) := (V, ρM , ρΦ , β) of the form ring R(mZII ) := (R, M, ψ, Φ = φ0 ) = (Z/mZ, Z/mZ, id, Z/2mZ) , where V = Z/mZ, ρM (a)(x, y) = 1 m axy, and ρΦ (φ0 )(x) := 1 2 x . 2m If C is a self-dual code of Type ρ(mZII ) then Sφ0 (C) is the usual shadow S(C) (Conway and Sloane [130], [454]). Remark. The above (by now classical) example illustrates the situation where shadows appear. Usually we begin with the smaller form ring. Given a ˜ := (R, M, ψ, Φ) ˜ (where the sub-form ring R := (R, M, ψ, Φ) of a form ring R ˜ ˜ ˜ strictly only diﬀerence between R and R is that the form structure Φ of R contains that of R) and a ﬁnite representation ρ of R that extends to a rep˜ (such that ρ is the restriction ρ˜|R ), then for any φ0 ∈ Φ˜ \ Φ resentation ρ˜ of R there is a φ0 -shadow of any code C of Type ρ, namely ρN Sφ0 (C) = {v ∈ VρN | β N (v, c) = (˜ Φ (φ0 ))(c) for c ∈ C} .

1)), where β(v, w) = 1≤i≤n ψ(1)(vi ⊗ wi ). 1. 3) of Matn (R, M, ψ, Φ) is deﬁned in the obvious way. That is, V n is a Matn (R)module via the usual matrix multiplication. The representation ρMatn (M ) is deﬁned by m11 . . m1n n .. . mij (xi , yj ) , ρMatn (M ) . . . (x, y) := mn1 . . mnn i,j=1 for all x = (x1 , . . , xn ) and y = (y1 , . . , yn ) ∈ V n , mij ∈ M (1 ≤ i, j ≤ n). The representation ρΦn is deﬁned by φ1 m12 . . m1n .. .. n . . (x) = ρΦn φi (xi ) + mij (xi , xj ) , ..

36 2 Weight Enumerators and Important Types Remark. H # is a subgroup of G which is canonically isomorphic to the character group of G/H. 2. Let G be a ﬁnite abelian group. The order of G is |G| = |G| . Moreover, the elements of G form an orthonormal basis for the space of all functions f : G → C with respect to the Hermitian inner product f1 , f2 G := 1 |G| f1 (g)f2 (g) . g∈G Remark. 2 is a standard result in representation theory (see for instance Serre [479]). e. those functions that are constant on the conjugacy classes of G.