By Elwyn R Berlekamp

This is often the revised version of Berlekamp's recognized booklet, "Algebraic Coding Theory", initially released in 1968, in which he brought a number of algorithms that have in this case ruled engineering perform during this box. this kind of is an set of rules for deciphering Reed-Solomon and Bose–Chaudhuri–Hocquenghem codes that accordingly 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 ordinary in symbolic manipulation structures. different novel algorithms superior the elemental equipment for doing numerous mathematics operations in finite fields of attribute . different significant examine contributions during this booklet integrated a brand new type of Lee metric codes, and distinctive asymptotic effects at the variety of details 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 publication to be of significant value.

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

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The relative complexity of the simplest known double-error-correcting binary codes contrasts sharply with the simplicity of the Hamming codes. 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.

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. 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.

A linear code may be decoded as follows. 1 Slepian's standard array Words Syndrome 000 001 010 011 100 101 110 111 000000 000001 000010 001000 000100 010000 100000 001100 001011 001010 001001 000011 001111 011011 101011 000111 010101 010100 010111 011101 010001 000101 110101 011001 011110 100110 101101 011111 100111 101100 011100 100100 101111 010110 101110 100101 011010 100010 101001 001110 110110 111101 111110 000110 001101 010010 101010 100001 110011 110010 110001 111011 110111 100011 010011 111111 111000 111001 111010 110000 111100 101000 011000 110100 page 7 March 3, 2015 6:6 Algebraic Coding Theory (Revised Edition) 9in x 6in b2064-ch01 ALGEBRAIC CODING THEORY 8 ing this syndrome, and subtract this coset leader from the received word to find a most likely transmitted codeword.