By L.R. Vermani
Coding conception got here into lifestyles within the past due 1940's and is anxious with devising effective encoding and deciphering procedures.
The publication is meant as a critical textual content for first classes in coding and algebraic coding idea, and is aimed toward complex undergraduates and up to date graduates as either a path and self-study textual content. BCH and cyclic, workforce codes, Hamming codes, polynomial in addition to many different codes are brought during this textbook. Incorporating a number of labored examples and entire logical proofs, it really is a terrific creation to the elemental of algebraic coding.
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Additional info for Elements of Algebraic Coding Theory
Choose a 0 of: f(X)ei such that degf(X) ~ degg(X)'v'O ¢ g(X)ei This choice is possible because the set of non-negative integers is well ordered. Let g(X)eF[X], g(X) ¢ 0. We claim that g(X) = q(X)f(X) + r(X) for some q(X), r(X)eF[X] where either r(X) = 0 or degr(X) < degf(X). If degg(X) < degf(X), we can take q(X) = 0 and r(X) = f(X). Suppose that n = degf(X) ~ degg(X) Let f(X) = a0 X" + a 1 X"- 1 + ... +an g(X) = b0 Xm + b1 Xm- 1 + ... + bm Then a0 of: 0 and g(X) = a(/b 0 Xm-nf(X) + g 1 (X) where deg g 1 (X) ~ m - 1.
3 In any finite field F = GF(pn), the multiplicative group F* of all non-zero elements is cyclic. Proof The multiplicative group F* of F is an Abelian group of order q - 1, where q = pn. If F* is not cyclic, there exists an integer r, 1 < r < q- 1 such that a'= 1'r/aeF*. Thus,everyaeFisarootofthepolynomialxr+l- X and hence X -aiX'+ 1 -XIrlaeF Also X- a, X- bare relatively coprime for a, beF, a =F b. Therefore n(x -a)IX'+ 1 -X But aeF deg and n (X- a)= O(F) = q aeF r+1 The number 32 + 8 + 4 + 1 = 45 in the denary system. In this way, it is quite easy to give denary representation of a number, binary representation of which is given. The reverse process is also equally simple and given any number in the denary system, we can get its binary representation. We explain this procedure through an example. 1 Consider the number 67. Observe that 67 = 64 + 2 + 1 = 26 + 2 1 + 1 and so binary representation ofthe number 67 is 1000011. Similarly 43 = 32 + 11 = 32 + 8 + 2 + 1 = 2 5 + 23 + 21 + 1 and its binary representation is 101011.
Elements of Algebraic Coding Theory by L.R. Vermani
The number 32 + 8 + 4 + 1 = 45 in the denary system. In this way, it is quite easy to give denary representation of a number, binary representation of which is given. The reverse process is also equally simple and given any number in the denary system, we can get its binary representation. We explain this procedure through an example. 1 Consider the number 67. Observe that 67 = 64 + 2 + 1 = 26 + 2 1 + 1 and so binary representation ofthe number 67 is 1000011. Similarly 43 = 32 + 11 = 32 + 8 + 2 + 1 = 2 5 + 23 + 21 + 1 and its binary representation is 101011.