By Richard E. Blahut

Error-correcting codes play a primary function in glossy communications and data-storage platforms. This quantity presents an obtainable advent to the elemental components of algebraic codes and discusses their use in a number of purposes. the writer describes a number vital coding concepts, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. in the course of the e-book, mathematical idea is illustrated by way of connection with many functional examples. The e-book is written for graduate scholars of electric and computing device engineering and practising engineers whose paintings contains communications or sign processing.

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0 . . 1 1 .. . .. . , , or a . . . 1 ... 0 a . . 1 . .. .. . 1 1 1 Each of these matrices has all remaining diagonal elements equal to one and all remaining nondiagonal elements equal to zero. Each matrix has an inverse matrix of a similar form. Elementary row operations are used to put a matrix in a standard form, called the row–echelon form, which is deﬁned as follows.

Calculate their rates. c. Write an expression for the probability of decoding error, pe , when the code is used with a binary channel that makes errors with probability q. How does the probability of error behave with n? 4. There is no need to show repetitive details (that is, show the principle). 5 For any (n, k) block code with minimum distance 2t + 1 or greater, the number of data symbols satisﬁes n − k ≥ logq 1 + n 1 (q − 1) + n 2 (q − 1)2 + · · · + n t (q − 1)t . Prove this statement, which is known as the Hamming bound.

The set is closed under multiplication, and the set of nonzero elements is an abelian group under multiplication. 3. The distributive law (a + b)c = ac + bc holds for all a, b, c in the set. It is conventional to denote by 0 the identity element under addition and to call it “zero;” to denote by −a the additive inverse of a; to denote by 1 the identity element under multiplication and to call it “one;” and to denote by a −1 the multiplicative inverse of a. By subtraction a − b, we mean a + (−b); by division a/b, we mean b−1 a.