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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.
The code-rate of a given ECC system is defined as the ratio between the number of information bits and the total number of bits (i.e., information plus redundancy bits) in a given communication package. The code-rate is hence a real number.
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.
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.
Fletcher addresses both of these weaknesses by computing a second value along with the simple checksum. This is the modular sum of the values taken by the simple checksum as each block of the data word is added to it.
The quantity is called the relative redundancy and gives the maximum possible data compression ratio, when expressed as the percentage by which a file size can be decreased. (When expressed as a ratio of original file size to compressed file size, the quantity R : r {\displaystyle R:r} gives the maximum compression ratio that can be achieved.)
The final digit of a Universal Product Code, International Article Number, Global Location Number or Global Trade Item Number is a check digit computed as follows: [3] [4]. Add the digits in the odd-numbered positions from the left (first, third, fifth, etc.—not including the check digit) together and multiply by three.
All Cyclic Redundancy Checks *are* calculated values using polynomials. Your statement accusing this article of not being necessarily true is "All CRCs are not calculated values (let alone polynomials)." However, the article is titled "Cyclic redundancy check", not "CRC".