By San Ling

ISBN-10: 0521529239

ISBN-13: 9780521529235

Eager about effectively transmitting facts via a loud channel, coding concept might be utilized to digital engineering and communications. in response to the authors' broad instructing adventure, this article offers a very smooth and obtainable direction at the topic. It comprises sections on linear programming and interpreting equipment crucial for modern arithmetic. quite a few examples and routines make the quantity excellent for college students and teachers.

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The general theory of finite fields began with the work of Carl Friedrich Gauss (1777–1855) and Evariste Galois (1811–1832), but it only became of interest for applied mathematicians and engineers in recent decades because of its many applications to mathematics, computer science and communication theory. Nowadays, the theory of finite fields has become very rich. In this chapter, we only study a small portion of this theory. The reader already familiar with the elementary properties of finite fields may wish to proceed directly to the next chapter.

Using these results, we are ready to prove the main characterization of finite fields. 3 For any prime p and integer n ≥ 1, there exists a unique finite field of p n elements. Proof. 28(ii) by showing that I p (n) > 0 for all primes p and integers n > 0). 6 that the residue ring Z p [x]/( f (x)) is in fact a field. It is easy to verify that this field has exactly p n elements. 3 Structure of finite fields 27 (Uniqueness) Let E and F be two fields of p n elements. In the composite n field E · F, consider the polynomial x p − x over E · F.

14 A finite field F of characteristic p contains p n elements for some integer n ≥ 1. Proof. Choose an element α1 from F ∗ . We claim that 0 · α1 , 1 · α1 , . . , ( p − 1) · α1 are pairwise distinct. Indeed, if i · α1 = j · α1 for some 0 ≤ i ≤ j ≤ p − 1, then ( j − i) · α1 = 0 and 0 ≤ j − i ≤ p − 1. , i = j. If F = {0 · α1 , 1 · α1 , . . , ( p − 1) · α1 }, we are done. Otherwise, we choose an element α2 in F\{0 · α1 , 1 · α1 , . . , ( p − 1) · α1 }. We claim that a1 α1 + a2 α2 are pairwise distinct for all 0 ≤ a1 , a2 ≤ p − 1.

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Coding Theory: A First Course by San Ling

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