By Jørn Justesen and Tom Høholdt

This publication is written as a textual content for a path geared toward complicated undergraduates. just some familiarity with undemanding linear algebra and likelihood is without delay assumed, yet a few adulthood is needed. the scholars could focus on discrete arithmetic, computing device technological know-how, or verbal exchange engineering. The e-book can also be an appropriate advent to coding idea for researchers from similar fields or for pros who are looking to complement their theoretical foundation. It provides the coding fundamentals for engaged on initiatives in any of the above components, yet fabric particular to 1 of those fields has now not been integrated. Chapters disguise the codes and interpreting equipment which are presently of so much curiosity in study, improvement, and alertness. they provide a comparatively short presentation of the fundamental effects, emphasizing the interrelations among diversified equipment and proofs of all vital effects. a series of difficulties on the finish of every bankruptcy serves to study the consequences and provides the scholar an appreciation of the innovations. moreover, a few difficulties and recommendations for tasks point out path for additional paintings. The presentation encourages using programming instruments for learning codes, imposing interpreting equipment, and simulating functionality. particular examples of programming workout are supplied at the book's domestic web page. disbursed in the Americas through the yankee Mathematical Society.

**Read or Download A Course in Error-Correcting Codes (EMS Textbooks in Mathematics) PDF**

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**Example text**

12 Factorize x 73 − 1 over F2 . 13 Factorize x 85 − 1 over F2 . 14 Factorize x 18 − 1 over F2 . 15 Is x 8 + x 7 + x 6 + x 5 + x 4 + x + 1 an irreducible polynomial over F2 ? 16 1) Show that f (x) = x 4 + x 3 + x 2 + x + 1 is irreducible in F2 [x]. 2) Construct F16 using f (x) , that is explain what the elements are and how to add and multiply them. 3) Determine a primitive element. 4) Show that the polynomial z 4 + z 3 + z 2 + z + 1 has four roots in F16 . 17 Let f (x) ∈ F2 [x] be an irreducible polynomial of degree m.

This is a non-systematic encoding; we will describe a systematic encoding in a problem in the next chapter. Since for Reed-Solomon Codes we must have n ≤ q there are no interesting binary codes. Codes over Fq where q is a prime make easier examples and in particular the field F11 is useful for decimal codes, since there is no field with ten elements. However in most practical cases we have q = 2m . 1. Reed-Solomon codes over F11 Since 2 is a primitive element of F11 we can take xi = 2i−1 mod 11, i = 1, 2, .

4) What are the possible orders of the elements? 5) Determine for all the possible orders an element of that order. 6) How many primitive elements are there?. 7) Try to solve the equation x 2 + x + 1 = 0. 8) Try to solve the equation x 2 + x − 6 = 0. 2 Let F be a field. 1) Show that a · b = 0 ⇒ a = 0 or b = 0. 2) Show that {0, 1, 2, 3} with addition and multiplication modulo 4 is not a field. 3 Let a ∈ Fq . 4 Determine all binary irreducible polynomials of degree 3. 5 Construct F8 using f (x) = x 3 + x + 1 , that is explain what the elements are and how to add and multiply them.