By San Ling

ISBN-10: 0521529239

ISBN-13: 9780521529235

All for effectively transmitting facts via a loud channel, coding concept will be utilized to digital engineering and communications. in response to the authors' wide educating adventure, this article offers a very sleek and available direction at the topic. It comprises sections on linear programming and deciphering equipment crucial for modern arithmetic. a variety of examples and routines make the amount perfect for college kids and teachers.

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**Example text**

An element of F[x] is called a n polynomial over F. 2 Polynomial rings 23 the degree of f (x), denoted by deg( f (x)), if an = 0 (for convenience, we n define deg(0) = −∞). Furthermore, a nonzero polynomial f (x) = i=0 ai x i of degree n is said to be monic if an = 1. A polynomial f (x) of positive degree is said to be reducible (over F) if there exist two polynomials g(x) and h(x) over F such that deg(g(x)) < deg( f (x)), deg(h(x)) < deg( f (x)) and f (x) = g(x)h(x). Otherwise, the polynomial f (x) of positive degree is said to be irreducible (over F).

2) that (a1 , a2 ) = (b1 , b2 ). As F has only finitely many elements, we can continue in this fashion and obtain elements α1 , . . , αn such that αi ∈ F\{a1 α1 + · · · + ai−1 αi−1 : a1 , . . , ai−1 ∈ Z p } for all 2 ≤ i ≤ n, and F = {a1 α1 + · · · + an αn : a1 , . . , an ∈ Z p }. In the same manner, we can show that a1 α1 + · · · + an αn are pairwise distinct for all ai ∈ Z p , i = 1, . . , n. This implies that |F| = p n . 1 Let F be a field. The set n ai x i : ai ∈ F, n ≥ 0 F[x] := i=0 is called the polynomial ring over F.

Ii) Let B = {v1 , . . , vk } denote a basis for V . Since v1 = 0, there are k q − 1 choices for v1 . For B to be a basis, the condition v2 ∈< v1 > is needed, so there are q k − q choices for v2 . Arguing in this manner, for every i such that k ≥ i ≥ 2, we need vi ∈< v1 , . . , vi−1 >, so there are q k − q i−1 choices k−1 k for vi . Hence, there are i=0 (q − q i ) distinct ordered k-tuples (v1 , . . , vk ). However, since the order of v1 , . . 1 i=0 qk − qi . 16 Let q = 2, S = {0001, 0010, 0100} and V =< S >, then V = {0000, 0001, 0010, 0100, 0011, 0101, 0110, 0111}.

### Coding Theory: A First Course by San Ling

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