By Richard E. Blahut
Error-correcting codes play a primary position in smooth communications and data-storage structures. This quantity presents an obtainable creation to the fundamental parts of algebraic codes and discusses their use in a number of functions. the writer describes a variety of very important coding thoughts, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. in the course of the ebook, mathematical conception is illustrated by means of connection with many useful examples. The publication is written for graduate scholars of electric and desktop engineering and working towards engineers whose paintings consists of communications or sign processing.
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Extra resources for Algebraic codes for data transmission
Let A0 = A. If Ar −1 contains ur , then let Ar = Ar −1 . Otherwise, ur does not appear in Ar −1 , but it can be expressed as a linear combination of the elements of Ar −1 , involving some element v j of A not in B. Form Ar from Ar −1 by replacing v j with ur . Any vector v is a linear combination of the elements of Ar −1 and so, too, of Ar because v j can be eliminated by using the linear equation that relates v j to ur and the other elements of Ar −1 . Therefore the set Ar spans V . From Ar −1 we have constructed Ar with the desired properties, so we can repeat the process to eventually construct Am , and the proof is complete.
One way of constructing groups with interesting algebraic structures is to study transformations of simple geometrical shapes and mimic these with the algebra. For example, an equilateral triangle with vertices A, B, and C (labeled clockwise) can be rotated or reﬂected into itself in exactly six different ways, and each of these has a rotation or reﬂection inverse. By making use of some obvious facts in this geometrical situation, we can quickly construct an algebraic group. Let the six transformations be denoted by the labels 1, a, b, c, d, and e as follows: 1 = (ABC → ABC) (no change) a = (ABC → BC A) (counterclockwise rotation) b = (ABC → C AB) (clockwise rotation) 2 In general the arithmetic operation in a group need not be commutative, and in that case is regarded as multiplication rather than addition.
Construct the array as follows: The ﬁrst row consists of the elements of H , with the identity element at the left and every other element of H appearing once and only once. Choose any element of G not appearing in the ﬁrst row. 2 Groups it as the ﬁrst element of the second row. The rest of the elements of the second row are obtained by multiplying each subgroup element by this ﬁrst element on the left. Continue in this way to construct a third, fourth, and subsequent rows, if possible, each time choosing a previously unused group element for the element in the ﬁrst column.