By Richard E. Blahut

Error-correcting codes play a basic function in smooth communications and data-storage platforms. This quantity offers an obtainable advent to the fundamental components of algebraic codes and discusses their use in a number of purposes. the writer describes a number of very important coding options, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. during the e-book, mathematical idea is illustrated via connection with many functional examples. The booklet is written for graduate scholars of electric and computing device engineering and practising engineers whose paintings consists of communications or sign processing.

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**Extra info for Algebraic codes for data transmission**

**Sample text**

1. Of course, the idea of the code is not changed if the bit positions are permuted. All of these variations are equivalent, and all are called the (7, 4) Hamming code. The decoder receives a seven-bit senseword v = (a0 , a1 , a2 , a3 , p0 , p1 , p2 ). This corresponds to a transmitted codeword with at most one error. 4(b), computes s0 = p0 + a0 + a1 + a2 s1 = p1 + a1 + a2 + a3 s2 = p2 + a0 + a1 + a3 . The three-bit pattern (s0 , s1 , s2 ) is called the syndrome. It does not depend on the actual data bits, but only on the error pattern.

One way to obtain a subgroup H of a ﬁnite group G is to take any element h from G, and let H be the set of elements obtained by multiplying h by itself an arbitrary number of times. That is, form the sequence of elements h, h ∗ h, h ∗ h ∗ h, h ∗ h ∗ h ∗ h, . . , denoting these elements more simply by h, h 2 , h 3 , h 4 , . . Because G is a ﬁnite group, only a ﬁnite number of these elements can be distinct, so the sequence must eventually repeat. The ﬁrst element repeated must be h itself, because if two other elements h i and h j are equal, they can be multiplied by the inverse of h, and thus h i−1 and h j−1 are also equal.

Gm ∗ h n . The ﬁrst element on the left of each row is known as a coset leader. Each row in the array is known as a left coset, or simply as a coset when the group is abelian. Alternatively, if the coset decomposition is deﬁned with the elements of G multiplied on the right, the rows of the array are known as right cosets. The coset decomposition is always rectangular, with all rows completed, because it is constructed that way. We shall prove that we always obtain an array in which every element of G appears exactly once.