By Richard E. Blahut

Error-correcting codes play a primary position in glossy communications and data-storage structures. This quantity presents an obtainable advent to the elemental parts of algebraic codes and discusses their use in a number of functions. the writer describes quite a number vital coding strategies, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. through the booklet, mathematical idea is illustrated by way of connection with many sensible examples. The publication is written for graduate scholars of electric and machine engineering and practising engineers whose paintings includes communications or sign processing.

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**Example text**

We then solve the set of equations by Cramer’s rule: x=D −1 1 1 0 1 1 0 1 1 −1 0 = 0, y = D 1 1 1 This is the same answer as before. 1 0 1 1 1 −1 0 = 0, z = D 1 1 1 1 1 0 1 0 = 1. 1. The ﬁeld G F(16) in hexadecimal notation A second example of a ﬁeld is known as G F(16). This ﬁeld has exactly sixteen elements, which can be denoted by the sixteen four-bit numbers, or can be denoted by the sixteen symbols of the hexadecimal alphabet {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}. 1. Notice that addition and multiplication are quite different from the familiar operations on the integers.

The group operation ∗ is taken to be this composition of permutations. For example, take n = 4. There are 4! = 24 permutations in S4 . A typical element of S4 is a = [(1234) → (3142)], 26 Introduction to Algebra which is the permutation that replaces 1 by 3, 2 by 1, 3 by 4, and 4 by 2. Another such permutation is b = [(1234) → (4132)]. Then in S4 , the product b ∗ a is the permutation obtained by applying ﬁrst a, then b. That is, b ∗ a = [(1234) → (2341)], which is an element of S4 . With this deﬁnition of multiplication, the permutation group S4 is a nonabelian group with twenty-four elements.

A group that consists of all the powers of one of its elements is called a cyclic group. Given a ﬁnite group G and a subgroup H , there is an important construction known as the coset decomposition of G, which illustrates certain relationships between H and G. Let the elements of H be denoted by h 1 , h 2 , h 3 , . . , and choose h 1 to be the identity element. 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.