By Richard E. Blahut

Error-correcting codes play a primary function in glossy communications and data-storage structures. This quantity offers an available creation to the fundamental components of algebraic codes and discusses their use in quite a few functions. the writer describes a variety of very important coding innovations, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. in the course of the publication, mathematical idea is illustrated via connection with many functional examples. The booklet is written for graduate scholars of electric and machine engineering and training engineers whose paintings contains communications or sign processing.

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8 a. Show that a code C is capable of detecting any pattern of d or fewer errors if and only if the minimum distance of the code C is greater than d. b. Show that a code is capable of correcting any pattern of t or fewer errors if and only if the minimum distance of the code is at least 2t + 1. c. Show that a code can be used to correct all patterns of t or fewer errors and, simultaneously, detect all patterns of d or fewer errors (d ≥ t) if the minimum distance of the code is at least t + d + 1.

A. Show that pk2 ,k1 is a check on both its column and its row. b. Show that this is a single-error-correcting code. c. Show that this code is also a double-error-detecting code. Give two doubleerror patterns that cannot be distinguished from one another when using this code and so cannot be corrected. d. What is the minimum distance of the code? 7 Show that Hamming distance has the following three properties: (i) d(x, y) ≥ 0 with equality if and only if x = y; (ii) d(x, y) = d(y, x). 19 Problems (iii) Triangle inequality d(x, y) ≤ d(x, z) + d(y, z).

3 The rows of an n by m matrix A over G F(q) may be thought of as a set of vectors of length m. 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.