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 to that end ruled engineering perform during this box. this type of is an set of rules for deciphering Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that thus grew to become referred to as 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 conventional in symbolic manipulation platforms. different novel algorithms enhanced the fundamental tools for doing a number of mathematics operations in finite fields of attribute . different significant study contributions during this publication integrated a brand new classification of Lee metric codes, and specific asymptotic effects at the variety of info symbols in lengthy binary BCH codes.

chosen chapters of the e-book grew to become a typical graduate textbook.

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

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

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**Extra resources for Algebraic Coding Theory**

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I Proof for polynomials Since fl JCi>(x)•; has no factors in common j r'i with j(i>(x)••, we may use Euclid's algorithm to find a polyno1 modj*(x)••. We then mial a<•>(x) such that a(i)(x) fl j (x)•; = j¢i set h(x) = 2: gCil(x)a(x) nf(j)(x)•;. i It is obvious that h(x) = j¢i g(i)(x) modf<"(x)•• for all i. (i)(x)••. D. The proof of the Chinese remainder theorem for integers is directly analogous to the above proof of the Chinese remainder theorem for polynomials. The consequences of Euclid's algorithm for polynomials are much the same as the consequences of Euclid's algorithm for integers. *

A single channel error (or, for that matter, any odd number of channel errors) will be detected as a decoding failure. Any combination of two channel errors (or, for that matter, any nonzero even number of channel errors) will cause a decoding error. These two examples, the repetition codes and the single-paritycheck codes, provide the extreme, relatively trivial, cases of binary codes. The repetition codes have enormous error-correction capability, but only one message digit per block. The single-parity-check codes have very high information rate, but since they contain only one check digit per block, they are unable to do more than detect an odd number of channel errors.

In fact, most of Hamming's results had been anticipated in a slightly different context by Fisher in 1942, in a paper which was well known to Bose! page 11 March 3, 2015 6:6 Algebraic Coding Theory (Revised Edition) u 9in x 6in b2064-ch01 ALGEBRAIC CODING THEORY In any case, the conceptual gap between the Hamming codes and the double-error-correcting BCH codes is considerable. For that reason, the student is urged to study the next section with care. In my opinion, it is the most important section of this book.