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

ISBN-10: 3540283714

ISBN-13: 9783540283713

This textual content bargains an intensive advent to the mathematical suggestions at the back of the idea of error-correcting linear codes. Care is taken to introduce the mandatory algebraic suggestions, for example the idea of finite fields, the polynomial earrings over such fields and the ever-present thought of team activities that permits the category of codes through isometry. The e-book presents in-depth assurance of vital subject matters like cyclic codes and the coding conception utilized in compact disc avid gamers.

The ultimate 4 chapters disguise complicated and algorithmic issues just like the type of linear codes by way of isometry, the enumeration of isometry periods, random iteration of codes, using lattice foundation aid to compute minimal distances, the specific development of codes with given parameters, in addition to the systematic evaluate of representatives of all isometry periods of codes. Up before, those complex issues have basically been lined in learn papers.

The current booklet presents entry to those effects at a degree that is appropriate for graduate scholars of arithmetic, machine technology and engineering in addition to for researchers.

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**Extra resources for Error-Correcting Linear Codes: Classification by Isometry and Applications**

**Example text**

Show that dist(C ⊥ ) ≥ 3. 4 Classiﬁcation by Isometry As we have seen, the coding theoretic properties of a code depend primarily on the Hamming distances between different codewords and between codewords and non-codewords. For example, the closest pair of codewords determines the error-correction rate of a code. Moreover, it may be that one code can be mapped onto another by means of a map which preserves the Hamming distances. 4 Classiﬁcation by Isometry 29 as the other, as far as error-correction is concerned.

In general, both the manufacturer number and the item reference number consist of 5 digits. This means that in this case a manufacturer can have up to 105 products. For that reason, those manufacturers which produce a smaller number of products get longer manufacturer codes. The check digit is the last number. All 13 digits x13 , . . , x1 are taken from the set {0, 1, . . , 9}. The check digit x1 is determined by the other digits such that ∑ i ≡1 mod 2 xi + 3 · ∑ xi ≡ 0 mod 10 i ≡0 mod 2 is satisfied.

Vn−1 ) · M( ϕ;π ) , where M( ϕ;π ) is the matrix whose k-th column is zero except for the (i, k)-entry which is ϕ(i ). Here i = π (k), so that ⎛ M( ϕ;π ) := ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ 0 ⎜ ⎜ ⎜ ⎜ ⎝ ... 0 k 0 .. 0 ϕ(i ) 0 .. 0 ⎞ 0 ... ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 0 ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ i = π ( k ). 32 1. Linear Codes Conversely, any linear mapping with e(i) −→ ϕ(π (i ))e(π (i)), for ϕ : n → F ∗q and π ∈ Sn , is a linear isometry. Moreover, linear isometries are invertible, and the composition of two of them is again a linear isometry.

### Error-Correcting Linear Codes: Classification by Isometry and Applications by Anton Betten, Michael Braun, Harald Fripertinger, Adalbert Kerber, Axel Kohnert, Alfred Wassermann

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