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So CRC method can be used to correct single-bit errors as well (within those limits, e.g. 32,767 bits with optimal generator polynomials of degree 16). Since all odd errors leave an odd residual, all even an even residual, 1-bit errors and 2-bit errors can be distinguished.
Even so, the Castagnoli CRC-32C polynomial used in iSCSI or SCTP matches its performance on messages from 58 bits to 131 kbits, and outperforms it in several size ranges including the two most common sizes of Internet packet. [16] The ITU-T G.hn standard also uses CRC-32C to detect errors in the payload (although it uses CRC-16-CCITT for PHY ...
To maximise computation speed, an intermediate remainder can be calculated by first computing the CRC of the message modulo a sparse polynomial which is a multiple of the CRC polynomial. For CRC-32, the polynomial x 123 + x 111 + x 92 + x 84 + x 64 + x 46 + x 23 + 1 has the property that its terms (feedback taps) are at least 8 positions apart ...
Over () (i.e. with binary BCH codes), this process is indistinguishable from appending a cyclic redundancy check, and if a systematic binary BCH code is used only for error-detection purposes, we see that BCH codes are just a generalization of the mathematics of cyclic redundancy checks.
By far the most popular FCS algorithm is a cyclic redundancy check (CRC), used in Ethernet and other IEEE 802 protocols with 32 bits, in X.25 with 16 or 32 bits, in HDLC with 16 or 32 bits, in Frame Relay with 16 bits, [3] in Point-to-Point Protocol (PPP) with 16 or 32 bits, and in other data link layer protocols.
It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a generator polynomial, which is used as the divisor in a polynomial long division over a finite field, taking the input data as the dividend. The remainder becomes the result. A CRC has properties that make it well suited for detecting burst ...
Cyclic redundancy checks (CRCs) can correct 1-bit errors for messages at most bits long for optimal generator polynomials of degree , see Mathematics of cyclic redundancy checks § Bitfilters; Locally Recoverable Codes
The cyclic redundancy check (CRC) is an error-detection code that operates by interpreting the message bitstring as the coefficients of a polynomial over GF(2) and dividing it by a fixed generator polynomial also over GF(2); see Mathematics of CRC.