Get Error-Correcting Linear Codes: Classification by Isometry PDF

By Anton Betten, Michael Braun, Harald Fripertinger, Adalbert Kerber, Axel Kohnert, Alfred Wassermann

This textual content bargains an creation to error-correcting linear codes for graduate scholars in arithmetic, computing device technology and engineering and researchers. The e-book differs from different average texts in its emphasis at the category of codes via isometry sessions. The suitable algebraic innovations like finite fields and crew activities are built carefully. Cyclic codes are mentioned in nice aspect, in addition to their software in CD avid gamers. within the final 4 chapters those isometry periods are enumerated, and representatives are built algorithmically without or with a prescribed automorphism workforce. moreover, lattice foundation aid is gifted as a device for computing generator matrices and the minimal distance of codes. The hooked up CD presents entry to generator matrices of greater than 70000 nonisometric optimum codes, masking all optimum codes for a given set of code parameters. It additionally comprises software program for comparing minimal distances, weight enumerators, and for the development of codes.

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B(k−1) ) and C = (c(0) , . . , c(n−1) ). Prove the following: Every vector space homomorphism ϕ : V → W is uniquely determined by its values on a basis of V. Assume that ϕ( b( i) ) = ∑ κij c( j), j∈n κij ∈ F , for i ∈ k. If we collect the elements κij in form of a matrix we obtain the matrix representation ⎛ ⎞ κ00 κ01 ... κ0,n−1 ⎜ κ κ11 ... κ1,n−1 ⎟ ⎜ ⎟ 10 ⎟ MB ,C ( ϕ) := (κij ) = ⎜ .. .. ⎜ ⎟ . ⎝ ⎠ . . κk−1,0 κk−1,1 . . κk−1,n−1 of ϕ with respect to the bases B and C . Conversely, prove that any k × n-matrix (κij ) over F determines a vector space homomorphism ϕ : V → W such that MB ,C ( ϕ) = (κij ).

That is, we study linear maps of H (n, q). Recall that any linear map is defined by the images of the unit vectors. e. ι(e(i) ) = κ j e( j) , for suitable j ∈ n, κ j ∈ F ∗q := F q \ {0} . Moreover, the sum of two different unit vectors is of weight 2, and so different unit vectors are mapped under ι to nonzero multiples of different unit vectors. Hence, there exists a unique permutation π in the symmetric group Sn and a unique mapping ϕ from n = {0, . . , n − 1} to F ∗q such that ι(e(i) ) = ϕ(π (i ))e(π (i)).

G −1 So, the wreath product H X G comes together with an action of G on X. It may happen that the group H acts on another set Y, say. In this case, we can define an action of H X G on the set of mappings Y X in the following way. H X G × Y X → Y X : (( ϕ; g), f ) → f , where f ( x ) := ϕ( x ) f ( g−1 x ). 9 This action is a host of further actions, some of which will be described next. These further actions are in fact actions of various subgroups of H X G (cf. 5). The first case is when the group G is trivial and all mappings ϕ : X → H are constant.

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