# Download e-book for kindle: Coding Theory: A First Course by San Ling 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.

Best mathematics books

New PDF release: Examples of the solutions of functional equations

Leopold vintage Library is thrilled to post this vintage ebook as a part of our wide assortment. As a part of our on-going dedication to offering worth to the reader, we now have additionally supplied you with a hyperlink to an internet site, the place you could obtain a electronic model of this paintings at no cost. the various books in our assortment were out of print for many years, and consequently haven't been available to most people.

Additional info for Coding Theory: A First Course

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}.