By Richard E. Blahut
Error-correcting codes play a basic position in smooth communications and data-storage platforms. This quantity offers an available advent to the fundamental parts of algebraic codes and discusses their use in various functions. the writer describes a number vital coding options, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. through the e-book, mathematical idea is illustrated through connection with many functional examples. The e-book is written for graduate scholars of electric and machine engineering and practising engineers whose paintings contains communications or sign processing.
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Extra resources for Algebraic codes for data transmission
The row space of A is the subspace of G F(q)m consisting of all linear combinations of the row vectors of A. The dimension of the row space is called the row rank. Similarly, the columns of A may be thought of as a set of vectors of length n. The column space of A is the subspace of G F(q)n consisting of all linear combinations of column vectors of A. The dimension of the column space is called the column rank. The set of vectors v such that AvT = 0 is called the null space of the matrix A. It is clear that the null space is a vector subspace of G F(q)m .
I n is a permutation of the integers 1, 2, . . in is ±1 according to whether the permutation is an even or odd permutation. An odd permutation is one that can be obtained as a product of an odd number of pairwise transpositions. An even permutation is one that cannot be obtained as a product of an odd number of pairwise transpositions. A transposition is an interchange of two terms. One way of visualizing the deﬁnition is to take the set of all matrices that can be obtained by permuting the rows of A.
First, we show that det(C) equals zero if either det(A) or det(B) equals zero. Suppose det(B) equals zero. 3(viii), the rows of B are linearly dependent. But the rows of C are linear combinations of the rows of B. Hence the rows of C are linearly dependent and det(C) equals zero. A similar argument is used if det(A) equals zero. Step 2. Suppose det(A) is not zero. Then it is possible to write A as a product of elementary matrices: A = F 1 F 2 · · · Fr . 3(iii), (v), and (vii), det(AB) = det[(F 1 F 2 · · · Fr )B] = det[F 1 (F 2 · · · Fr B)] = (det F 1 ) det(F 2 · · · Fr B) = (det F 1 )(det F 2 ) · · · (det Fr )(det B).