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The cyclic redundancy check (CRC) is a check of the remainder after division in the ring of polynomials over GF(2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around.
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.
As an example of implementing polynomial division in hardware, suppose that we are trying to compute an 8-bit CRC of an 8-bit message made of the ASCII character "W", which is binary 01010111 2, decimal 87 10, or hexadecimal 57 16.
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. [1] [6] After these discussions of specific codes, the next chapter concerns enumerator polynomials, including the MacWilliams identities, Pless's own power moment identities, and the Gleason ...
Given a prime number q and prime power q m with positive integers m and d such that d ≤ q m − 1, a primitive narrow-sense BCH code over the finite field (or Galois field) GF(q) with code length n = q m − 1 and distance at least d is constructed by the following method.
A quasi-cyclic code has the property that for some s, any cyclic shift of a codeword by s places is again a codeword. [9] A double circulant code is a quasi-cyclic code of even length with s=2. [9] Quasi-twisted codes and multi-twisted codes are further generalizations of constacyclic codes. [10] [11]
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.
Proof. We need to prove that if you add a burst of length to a codeword (i.e. to a polynomial that is divisible by ()), then the result is not going to be a codeword (i.e. the corresponding polynomial is not divisible by ()).