By Richard E. Blahut
Error-correcting codes play a primary position in smooth communications and data-storage structures. This quantity offers an available advent to the fundamental parts of algebraic codes and discusses their use in quite a few purposes. the writer describes quite a number very important coding thoughts, together with Reed-Solomon codes, BCH codes, trellis codes, and turbocodes. in the course of the publication, mathematical conception is illustrated via connection with many useful examples. The ebook is written for graduate scholars of electric and desktop engineering and training engineers whose paintings consists of communications or sign processing.
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Extra info for Algebraic codes for data transmission
Let the six transformations be denoted by the labels 1, a, b, c, d, and e as follows: 1 = (ABC → ABC) (no change) a = (ABC → BC A) (counterclockwise rotation) b = (ABC → C AB) (clockwise rotation) 2 In general the arithmetic operation in a group need not be commutative, and in that case is regarded as multiplication rather than addition. In a concrete case, even a commutative group operation might be called multiplication. 2 Groups c = (ABC → AC B) (reﬂection about bisector of angle A) d = (ABC → C B A) (reﬂection about bisector of angle B) e = (ABC → B AC) (reﬂection about bisector of angle C), where the transformation (ABC → BC A) means that vertex A goes into vertex B, vertex B goes into vertex C, and vertex C goes into vertex A.
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). A distance function with these three properties is called a metric.
Let G be a group and let H be a subset of G. Then H is called a subgroup of G if H is a group with respect to the restriction of ∗ to H . To prove that a nonempty set H is a subgroup of G, it is only necessary to check that a ∗ b is in H whenever a and b are in H (closure), and that the inverse of each a in H is also in H . The other properties required of a group then will be inherited from the group G. If the group is ﬁnite, then even the inverse property is satisﬁed automatically if the closure property is satisﬁed, as we shall see shortly in the discussion of cyclic subgroups.
Algebraic codes for data transmission by Richard E. Blahut