# Design of an object oriented Finite Element package for by Lingen F.

By Lingen F.

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3. Let C[V N ] denote the group ring of V N , that is, the vector space generated over C by the symbols ev with v ∈ V N . The full weight enumerator of C is ec ∈ C[V N ] . 4. ,vN ) → i=1 xvi . It is sometimes useful to symmetrize cwe(C) in other ways, by identifying the variables xa under the action of various permutation groups. If R = V = Fq is a ﬁnite ﬁeld, then symmetrizing cwe(C) by the Galois group of Fq over its prime ﬁeld yields the symmetrized weight enumerator swe(C). In the case R = V = Z/mZ, we may also identify xa and xb if a ∈ V and b ∈ V have the same Lee weight: this leads to the Lee-symmetrized weight enumerator, which we also denote by swe(C).

Therefore the automorphism group of ρ(q1E ) consists of the elements g ∈ Aut(q E ) that additionally satisfy Tr(gx) = Tr(x) for all x ∈ Fq . This implies that g = 1 and hence Aut(ρ(q1E )) = {1}. Therefore the ρ(q1E )-symmetrized weight enumerator is the same as the complete weight enumerator, a polynomial in q variables. Additional information about a code can be obtained by considering more than one codeword at a time, leading to what are called higher-genus or multiple weight enumerators. 7. Let c(i) := (c1 , .

Mn−1,n + mn,n−1  mnn The map λ : Φn → Matn (F2 ) is deﬁned by    0 φ1 m12 . . m1n    . . ..     =  m12 λ   .   . m  ..   n−1,n φn m1n m12 .. .. .. .. . . mn−1,n  m1n  ..  . ,  mn−1,n  0 and its image is the set of all symmetric matrices in Matn (F2 ) whose diagonal entries are 0. The latter is sometimes also denoted by Altn (F2 ), the set of all alternating matrices. The involution τ is given by transposition. 1). They arise naturally as the image of λn in the matrix ring of a form ring, if the image of λ is 2M .