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A Course in Error-Correcting Codes (EMS Textbooks in by Jørn Justesen and Tom Høholdt

By Jørn Justesen and Tom Høholdt

This publication is written as a textual content for a path geared toward complex undergraduates. just some familiarity with straight forward linear algebra and likelihood is at once assumed, yet a few adulthood is needed. the scholars might focus on discrete arithmetic, computing device technological know-how, or verbal exchange engineering. The ebook can also be an appropriate advent to coding conception for researchers from comparable fields or for pros who are looking to complement their theoretical foundation. It supplies the coding fundamentals for engaged on initiatives in any of the above components, yet fabric particular to at least one of those fields has no longer been integrated. Chapters hide the codes and interpreting tools which are at the moment of so much curiosity in examine, improvement, and alertness. they provide a comparatively short presentation of the fundamental effects, emphasizing the interrelations among various tools and proofs of all vital effects. a series of difficulties on the finish of every bankruptcy serves to study the consequences and provides the coed an appreciation of the innovations. additionally, a few difficulties and proposals for initiatives point out path for additional paintings. The presentation encourages using programming instruments for learning codes, imposing deciphering tools, and simulating functionality. particular examples of programming workout are supplied at the book's domestic web page. dispensed in the Americas by means of the yank Mathematical Society.

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1. 1 and suppose we receive r = (5, 9, 0, 9, 0, 1, 0, 7, 5). We have l0 = 7 and l1 = 3 and therefore we get 10 equations with 11 unknowns. The matrix becomes   1 1 1 1 1 1 1 1 5 5 5 5  1 2 4 8 5 10 9 7 9 7 3 6     1 4 5 9 3 1 4 5 0 0 0 0     1 8 9 6 4 10 3 2 9 6 4 10     1 5 3 4 9 1 5 3 0 0 0 0     1 10 1 10 1 10 1 10 1 10 1 10     1 9 4 3 5 1 9 4 0 0 0 0     1 7 5 2 3 10 4 6 7 5 2 3     1 3 9 5 4 1 3 9 0 0 0 0  1 6 3 7 9 10 5 8 5 8 4 2 The system has as a solution (4, 1, 2, 2, 2, 9, 1, 0, 7, 3, 10, 0) corresponding to Q 0 (x) = x 6 + 9x 5 + 2x 4 + 2x 3 + 2x 2 + x + 4 and Q 1 (x) = 10x 2 +3x +7.

The finite field F3 . If we let p = 3 we get the ternary field with elements 0, 1, 2. ) We will now prove that multiplication in any finite field F with q elements can essentially be done as addition of integers modulo q − 1. This results from the fact that F contains an element α, a so-called primitive element, such that F\{0} = {α i |i = 0, 1, . . , q − 2} and α q−1 = 1 . Therefore α i · α j = α (i+ j ) mod (q−1) . 2. Let F be a finite field with q elements, and let a ∈ F\{0}. The order of a, ord (a), is the smallest positive integer s such that a s = 1.

1 therefore m m α 2 = α and hence α 2 −1 = 1. 2 d|m and we are finished. 2. Let n be an odd number and let j be an integer 0 ≤ j < n. The cyclotomic coset containing j is defined as { j, 2 j mod n, . . , 2i j mod n, . . , 2s−1 j mod n} where s is the smallest positive integer such that 2s j mod n = j . If we look at the numbers, 2i j mod n, i = 0, 1, . . , they can not all be different so indeed the s in the definition above exists. If n = 15 we get the following cyclotomic cosets: {0} {1, 2, 4, 8} {3, 6, 12, 9} {5, 10} {7, 14, 13, 11} We will use the notation that if j is the smallest number in a cyclotomic coset, then that coset is called C j .

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