By Elwyn R Berlekamp
This is often the revised version of Berlekamp's recognized e-book, "Algebraic Coding Theory", initially released in 1968, in which he brought numerous algorithms that have for that reason ruled engineering perform during this box. the sort of is an set of rules for deciphering Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that for that reason 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 elaborations grew to become well-known in symbolic manipulation platforms. different novel algorithms superior the elemental tools for doing a variety of mathematics operations in finite fields of attribute . different significant examine contributions during this e-book integrated a brand new type of Lee metric codes, and specified asymptotic effects at the variety of info symbols in lengthy binary BCH codes.
chosen chapters of the publication 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 conception, and software program engineering.
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Additional info for Algebraic Coding Theory
For example, if S1 = 0 but Sa :¢ 0, the decoding algorithm fails, detecting three or more errors. In Chap. 481. The choice of fW = ~a is thus seen to provide a suitable method of choosing the lower five rows of the X matrix in order to yield a doubleerror-correcting binary code of block length 31, having 10 parity checks. The first five parity checks then give the sum of the error location numbers; the second five parity checks give the sum of the cubes of the error location numbers. The decoding procedure consists of three major steps: (1) compute the parity checks, obtaining S1 and Sa; (2) compute the error polynomial u(z); (3) find the reciprocals of the roots of u(z) and change the corresponding positions of the received word.
If exactly two errors were made, their location numbers both satisfy this equation. Since this equation then has two roots among the field of binary polynomials mod M(x), the decoder will always be able to find the two error location numbers. If there was only one error, then Hence, in this case the lone error satisfies the equation or Finally, the decoder can always decode if there were no errors, because in that case For reasons which will become apparent later (Sec. , multiplicative inverses, of the locations of the errors, rather than the locations of the errors themselves.
3 HAMMING CODESt At extremely low rates or extremely high rates, it is relatively easy to find good linear codes. In order to interpolate between these two extremes, we might adopt either of two approaches: (1) start with the low-rate codes and gradually increase k by adding more and more codewords, attempting to maintain a large error-correction capability, or (2) start with good high-rate codes and gradually increase the errorcorrection capability, attempting to add only a few additional paritycheck constraints.
Algebraic Coding Theory by Elwyn R Berlekamp