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By Ageev A.A., Baburin A.E., Gimandi E.K.

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Extra resources for A 3/4-Approximation Algorithm for Multiple Subset Sum

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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 . 1 30 1. Linear Codes Obvious isometries are the permutations of the coordinates. These isometries will be called permutational isometries.

Codes over the field F 2 := {0, 1} of two elements are called binary codes. Codes over the field F 3 := {0, 1, 2} of three elements are called ternary codes, whereas codes over a four-element field F 4 are called quaternary. 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 + .

Linear Codes For v ∈ F k , w ∈ F n and a generator matrix Γ of C it follows from v · Γ, w = v, w · Γ that C ⊥ = { w ∈ F n | w · Γ = 0} . This shows that the generator matrix Γ of C is a check matrix of C ⊥ . Consequently, C ⊥ is a linear (n, n − k)-code. Since (C ⊥ )⊥ = C (cf. 4 Corollary The check matrices of a code C are the generator matrices of the dual code C ⊥ and vice versa. Dually, the check matrices of the dual code are the generator ✷ matrices of the code. It is now time to present an example of a linear code which can correct one error.

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A 3/4-Approximation Algorithm for Multiple Subset Sum by Ageev A.A., Baburin A.E., Gimandi E.K.


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