Information Theory

Algebraic Coding Theory by Elwyn R Berlekamp

By Elwyn R Berlekamp

This is often the revised variation of Berlekamp's well-known booklet, "Algebraic Coding Theory", initially released in 1968, in which he brought a number of algorithms that have for that reason ruled engineering perform during this box. this type of is an set of rules for interpreting Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that hence 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 established in symbolic manipulation platforms. different novel algorithms enhanced the elemental equipment for doing a variety of mathematics operations in finite fields of attribute . different significant learn contributions during this booklet integrated a brand new type of Lee metric codes, and certain asymptotic effects at the variety of details symbols in lengthy binary BCH codes.

chosen chapters of the booklet grew to become a customary graduate textbook.

either working towards engineers and students will locate this e-book to be of serious value.

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

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For this reason, the continued-fractions version again provides an attractive alternative. Defining 0 = 1 1 =0 p(-2) = p(-1) = q(-2) q(-1) we compute p = ap = aq < deg r < deg r and r<- 1> must also divide r<- 1>p<"- 1> and r<- 2>q and hence that any common factor of r<- 2> and r<- 1> must also divide r.

4 MANIPULATIVE INTRODUCTION TO DOUBLE-ERROR-CORRECTING BCH CODES We have seen that a linear code is characterized by its parity-check matrix 3C. We have also seen that the syndrome of the received sequence is the sum of the columns of 3C corresponding to the error positions. Hence, a linear code is capable of correcting all single-error patterns iff all columns of 3C are different and nonzero. If 3C has m rows and can correct single errors, then n 2m - 1. The Hamming codes achieve this bound. Each digit of a Hamming code may be labeled by a nonzero binary m-tuple, which is equal to the corresponding column of the 3C matrix.

Then 3CR1 = 0, and 3CE1 = 0, so that the error pattern must also be a codeword. The set of n-dimensional binary vectors which have the zero syndrome is precisely the set of codewords. If x and y are two codewords, then 3Cx1 = 3Cy1 = 0, so 3Cx1 - 3Cy1 = 0, and x - y is also a codeword. Thus, the difference of any pair of codewords is another codeword. For this reason, the set of codewords is said to form a linear code, or a group code. More generally, if x andy have the same syndrome (not necessarily zero), then 3Cx' = 3Cy', 3C(x1 - y1) = 0, and x- y is a codeword.

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