By Elwyn R Berlekamp

This is often the revised variation of Berlekamp's recognized booklet, "Algebraic Coding Theory", initially released in 1968, in which he brought a number of algorithms that have consequently ruled engineering perform during this box. this type of is an set of rules for deciphering 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 turned frequent in symbolic manipulation platforms. different novel algorithms more desirable the fundamental equipment for doing quite a few mathematics operations in finite fields of attribute . different significant examine contributions during this e-book integrated a brand new category of Lee metric codes, and targeted asymptotic effects at the variety of info symbols in lengthy binary BCH codes.

chosen chapters of the booklet turned 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 concept, and software program engineering.

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**Extra resources for Algebraic Coding Theory**

**Example text**

These information digits are added according to the binary rules: 0 + 0 = 0, 0 + 1 = 1, 1 + 0 = 1, 1 + 1 = 0. The binary sum of a number of binary digits is seen to be 0 or 1, accordingly as the number of ones among these digits is even or odd. From this it follows that the total number of ones (including the check digit) in every codeword of a single-parity-check code is even. If the received word contains an even number of ones, the decoder may decode it without change, but if the received word page 3 March 3, 2015 6:6 Algebraic Coding Theory (Revised Edition) 4 9in x 6in b2064-ch01 ALGEBRAIC CODING THEORY contains an odd number of ones, the decoder should not decode it.

Each J

The last five rows of this matrix will form a Hamming code iff the function f is a permutation. Heuristically, this prospect seems hopeful. If either the first 1n checks or the second 1n checks can correct a single error, then perhaps somehow they can jointly correct two errors. In order to investigate the possible choices of our function j, we shall have to develop some facility for manipulating binary 5-tuples. We would like to be able to add, subtract, multiply, and divide 5-tuples, in some sense.