 By Christophe Dabancourt

ISBN-10: 2212123507

ISBN-13: 9782212123500

Apprendre à programmer : Algorithmes et notion objet

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Wn−1 ) = (v0 , . . , vk−1 ) · (κij ). This shows that a generator matrix Γ of an (n, k)-code over F describes a vector space homomorphism ϕ : F k → F n . In particular, if B = (b(0) , . . , b(k−1) ) is a basis of V, every endomorphism of V can be represented as a k × k-matrix over F with respect to this basis. Exercise Let V and W be two finite dimensional vector spaces over F of dimension k and n respectively. Show that a homomorphism ϕ : V → W is injective if and only if dim( ϕ(V )) = dim(V ).

2 Example Consider the following check matrix over the field F 2 = {0, 1} of two elements, consisting of a single row of length n ≥ 2, ∆ := 1 1 ... 1 . It is a check matrix of a binary (n, n − 1)-code C. Each codeword c = ( c0 , . . , c n −1 ) ∈ C is of even weight, since 0 = c · ∆ = c0 + . . + cn−1 ≡ wt(c) mod 2. , C consists of the vectors of even weight in F 2n , and so C has minimum distance d = 2. This shows that C can detect one error. It is called a parity check code, since C can be obtained in the following way: Take C := F 2n−1 as the message space and add to each of its elements c = (c0 , .

The equivalence class of x ∈ X with respect to R is the set [ x ] R := {y ∈ X | ( x, y) ∈ R} , and the set of all equivalence classes with respect to R is indicated as X/R. It forms a decomposition of X into pairwise disjoint and nonempty subsets. Instead of ( x, y) ∈ R we usually write x ∼ y where ∼ denotes the equivalence relation. e. d(w, w ) = d(ι(w), ι(w )), for all w, w ∈ H (n, q). Mappings with the latter property are called isometries. Using this notion we introduce the following concept which is in fact the central concept of the present book: Deﬁnition (isometric codes) Two linear codes C, C ⊆ H (n, q) are called isometric if there exists an isometry of H (n, q) that maps C onto C .