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Chapter 3 presents the BCH code over the field (), and Chapter 4 develops the theory of finite fields more generally. [1] [6] Chapter 5 studies cyclic codes and Chapter 6 studies a special case of cyclic codes, the quadratic residue codes. Chapter 7 returns to BCH codes.
In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called a Galois field). BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem, and independently in 1960 by Raj Chandra Bose and D. K. Ray ...
To convert (,) cyclic code to (,) shortened code, set symbols to zero and drop them from each codeword. Any cyclic code can be converted to quasi-cyclic codes by dropping every th symbol where is a factor of . If the dropped symbols are not check symbols then this cyclic code is also a shortened code.
A cyclic redundancy check (CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital data. [ 1 ] [ 2 ] Blocks of data entering these systems get a short check value attached, based on the remainder of a polynomial division of their contents.
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.
In coding theory, burst error-correcting codes employ methods of correcting burst errors, ... Theorem (Cyclic burst correction capability) ...
Computation of a cyclic redundancy check is derived from the mathematics of polynomial division, modulo two. In practice, it resembles long division of the binary message string, with a fixed number of zeroes appended, by the "generator polynomial" string except that exclusive or operations replace subtractions.
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.