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All error-detection and correction schemes add some redundancy (i.e., some extra data) to a message, which receivers can use to check consistency of the delivered message and to recover data that has been determined to be corrupted.
This book is mainly centered around algebraic and combinatorial techniques for designing and using error-correcting linear block codes. [ 1 ] [ 3 ] [ 9 ] It differs from previous works in this area in its reduction of each result to its mathematical foundations, and its clear exposition of the results follow from these foundations.
It is an error-correcting code capable of correcting up to three errors in each 24-bit word, and detecting a fourth. Richard Hamming won the Turing Award in 1968 for his work at Bell Labs in numerical methods, automatic coding systems, and error-detecting and
Low-density parity-check (LDPC) codes are a class of highly efficient linear block codes made from many single parity check (SPC) codes. They can provide performance very close to the channel capacity (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length.
The distance d was usually understood to limit the error-correction capability to ⌊(d−1) / 2⌋. The Reed–Solomon code achieves this bound with equality, and can thus correct up to ⌊(n−k) / 2⌋ errors. However, this error-correction bound is not exact.
CRCs can also be used as part of error-correcting codes, which allow not only the detection of transmission errors, but the reconstruction of the correct message. These codes are based on closely related mathematical principles.
To convolutionally encode data, start with k memory registers, each holding one input bit.Unless otherwise specified, all memory registers start with a value of 0. The encoder has n modulo-2 adders (a modulo 2 adder can be implemented with a single Boolean XOR gate, where the logic is: 0+0 = 0, 0+1 = 1, 1+0 = 1, 1+1 = 0), and n generator polynomials — one for each adder (see figure below).
Verhoeff had the goal of finding a decimal code—one where the check digit is a single decimal digit—which detected all single-digit errors and all transpositions of adjacent digits. At the time, supposed proofs of the nonexistence [ 6 ] of these codes made base-11 codes popular, for example in the ISBN check digit .