By Elwyn R Berlekamp

This can be the revised version of Berlekamp's recognized e-book, "Algebraic Coding Theory", initially released in 1968, in which he brought numerous algorithms that have accordingly ruled engineering perform during this box. this type of is an set of rules for interpreting Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that to that end grew to become 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 gildings turned normal in symbolic manipulation structures. different novel algorithms more desirable the elemental tools for doing a number of mathematics operations in finite fields of attribute . different significant learn contributions during this ebook integrated a brand new type of Lee metric codes, and distinctive asymptotic effects at the variety of info symbols in lengthy binary BCH codes.

chosen chapters of the booklet turned a typical graduate textbook.

either training engineers and students will locate this booklet to be of significant value.

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

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This can be the revised variation of Berlekamp's recognized e-book, "Algebraic Coding Theory", initially released in 1968, in which he brought numerous algorithms that have therefore ruled engineering perform during this box. this type of is an set of rules for interpreting Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that therefore turned referred to as the Berlekamp–Massey set of rules.

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A great deal of work in constructive coding theory followed the appearance of Hamming's pioneering paper in 1950. 1\Iany new types of codes were introduced, but, with a few important exceptions, most of the new constructions were specialized codes for very specialized purposes. Even the next simplest problem, the construction of relatively high-rate binary codes which would correct any pattern of double errors, remained unsolved. When double-error-correcting codes finally were discovered by Bose and Chaudhuri in 1960 and Hocquenghem in 1959, the generalization to t-error-correcting codes followed immediately, for all t.

It is further evident that a polynomial of degree d cannot have more than d linear factors. Furthermore, we claim that the linear polynomial x - ~is a factor of the polynomial f(x) iff fW = 0, that is, iff ~ is a root of the polynomial f(x). For if we apply the division algorithm to the dividend f(x), using x - ~for the divisor, we shall obtain a quotient q(x) and a remainder r. Since the remainder will have degree smaller than the divisor, the remainder must be a constant, independent of x. 15.

The set of all vectors having the same syndrome is said to be a coset of the group of codewords. As we have just shown, the difference between any two vectors in the same coset is a codeword. Conversely, the sum of some vector x and any codeword gives another vector in the same coset as x. Thus, the number of vectors in any coset is identical to the number of codewords, which is 2k. The possible error patterns are the words in the same coset as the received word. , error patterns which have different syndromes.