Self-Dual Codes and Invariant Theory (Algorithms and by Gabriele Nebe;Eric M. Rains;Neil J. A. Sloane

By Gabriele Nebe;Eric M. Rains;Neil J. A. Sloane

Essentially the most awesome and gorgeous theorems in coding thought is Gleason's 1970 theorem in regards to the weight enumerators of self-dual codes and their connections with invariant concept, which has encouraged hundreds and hundreds of papers approximately generalizations and functions of this theorem to sorts of codes. This self-contained booklet develops a brand new concept that's strong sufficient to incorporate all of the previous generalizations.

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Additional resources for Self-Dual Codes and Invariant Theory (Algorithms and Computation in Mathematics)

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If C is a singly-even self-dual code then C0 is the doubly-even index 2 subcode of C, and the shadow S(C) = C0⊥ \ C. 8 of Chapter 2 we shall not discuss it here. 5. There is a second representation of the form ring R(2II ) which is of interest. This is the representation ρ(2II ) := (F4 , ρM , ρΦ , β ), where β (x, y) = ρM (1)(x, y) = 1 1 x¯ y , ρΦ (φ0 )(x) = x¯ x, 2 2 and x → x denotes the nontrivial Galois automorphism of F4 . The selforthogonal codes in N ρ(2II ) are the additive trace-Hermitian self-orthogonal codes over the alphabet F4 .

M − 1} (which include codes over all finite fields of prime order) we define the Lee weight and Euclidean norm of u ∈ V by Lee(u) = min{u, m − u} , Norm(u) = (Lee(u))2 . For a vector u = (u1 , . . , uN ) ∈ V N , we set N Lee(u) = Lee(ui ) , i=1 N Norm(u) = Norm(ui ) . i=1 Of course, if u is a binary vector, wt(u) = Lee(u) = Norm(u). For example, for the ring Z/4Z = {0, 1, 2, 3}, the Lee weights are respectively {0, 1, 2, 1}. By analogy, Gaborit, Pless, Sol´e and Atkin [186] define “Lee” weights on the field F4 = {0, 1, ω, ω 2 } to be {0, 2, 1, 1}, respectively.

2. The properties of being self-orthogonal, self-dual and isotropic are preserved under permutation equivalence, equivalence and weak equivalence. 12 of Chapter 11. This section translates this notion into our new language of form rings and so opens up the possibility of generalizing the shadow methods to new Types of codes. Let ρ := (V, ρM , ρΦ , β) be a finite representation of the form ring (R, M, ψ, Φ). Let C be a self-orthogonal code in N ρ. 6). 1. The maximal isotropic subcode C0 of C is C0 := C ∩ ker(ρN Φ (φ)) .

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