By Richard E. Blahut
Error-correcting codes play a primary position in sleek communications and data-storage platforms. This quantity presents an available creation to the elemental components of algebraic codes and discusses their use in numerous purposes. the writer describes various very important coding options, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. through the ebook, mathematical idea is illustrated via connection with many functional examples. The ebook 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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Additional info for Algebraic Codes for Data Transmission
10 a. Show that if the binary (15, 11) Hamming code is used to correct single errors on a channel that makes two errors, the decoder output is always wrong. b. Show that if the two errors are in check bits, the decoder will always miscorrect a data bit. c. By appending an overall check bit, show how to extend the (15, 11) Hamming code to a (16, 11) code that corrects all single errors and detects all double errors. What is the minimum distance of this code? 1 is unchanged by the permutation (c0 , c1 , c2 , c3 , c4 , c5 , c6 ) → (c0 , c4 , c1 , c5 , c2 , c6 , c3 ).
There are eight possible syndromes: one that corresponds to no error, and one for each of the seven possible patterns with a single error. 2. It is a simple matter to design binary logic that will complement the bit location indicated by the syndrome. After correction is complete, the check bits can be discarded. 2. Syndrome table Syndrome 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 Error 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 If two or more errors occur, then the design speciﬁcation of the code is exceeded and the code will miscorrect.
The concept of an n-dimensional vector space is closely related to the ideas of linear algebra and matrix theory, and is important in many applications. Vector spaces also can be deﬁned abstractly with respect to any ﬁeld. 1. Let F be a ﬁeld. The elements of F will be called scalars. A set V is called a vector space, and its elements are called vectors, if there is deﬁned an operation called vector addition (denoted by +) on pairs of elements from V , and an operation called scalar multiplication (denoted by juxtaposition) on an element of F and an element of V to produce an element of V provided the following axioms are satisﬁed.