By Elwyn R Berlekamp

This is often the revised variation of Berlekamp's well-known e-book, "Algebraic Coding Theory", initially released in 1968, in which he brought numerous algorithms that have as a consequence ruled engineering perform during this box. this kind of is an set of rules for interpreting Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that in this case turned often called the Berlekamp–Massey set of rules. one other is the Berlekamp set of rules for factoring polynomials over finite fields, whose later extensions and elaborations grew to become wide-spread in symbolic manipulation platforms. different novel algorithms more advantageous the elemental equipment for doing a number of mathematics operations in finite fields of attribute . different significant study contributions during this e-book incorporated a brand new type of Lee metric codes, and exact asymptotic effects at the variety of info symbols in lengthy binary BCH codes.

chosen chapters of the booklet turned a customary graduate textbook.

either working towards engineers and students will locate this ebook to be of serious value.

Readership: Researchers in coding idea and cryptography, algebra and quantity concept, and software program engineering.

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A further study of fields will be undertaken in Chap. 4. page 15 March 3, 2015 6:6 Algebraic Coding Theory (Revised Edition) 9in x 6in b2064-ch01 ALGEBRAIC CODING THEORY 16 = = s(x)d(x), then a(x)s(x) = r(x)M(x) 0 mod M(x). If it were possible to divide by a(x), we could then deduce that s(x) = 0 mod M(x), which is not true. However, if (a(x),b(x)) = 1, then, according to Euclid, we can find A(x) and B(x) so that M(x) a(x)A(x) + b(x)B(x) = 1 = Whence a(x)A(x) 1 mod b(x). Evidently division by a(x) is equivalent to multiplication by A(x).

In particular, it is not always legitimate to divide by a(x) mod 11J(x) if a(x) and M(x) have a common factor d(x). " t A field is a set of elements, including 0 and 1, any pair of which may be added, subtracted, multiplied, or divided (except by 0) to give a unique result in the field. A further study of fields will be undertaken in Chap. 4. page 15 March 3, 2015 6:6 Algebraic Coding Theory (Revised Edition) 9in x 6in b2064-ch01 ALGEBRAIC CODING THEORY 16 = = s(x)d(x), then a(x)s(x) = r(x)M(x) 0 mod M(x).

The matrix JC is called the parity-check matrix. The code's block length n is equal to the number of columns of :JC. The syndrome s of any word R is defined by the equation s 1 = :JCR1• A coset consists of all the words having a given syndrome. The weight of any word is the number of ones among its n digits. Within each coset, a word having the least weight is chosen as coset leader. 22. 22 If R is the received word, the set of possible error words is the coset containing R. A most probable error word is the leader of the coset containing R.