CRCcodes

CRC codes, or cyclic redundancy check codes, are error-detecting codes used to detect accidental data corruption in digital communications and storage. They work by treating the message as a polynomial over the binary field GF(2) and dividing it by a fixed generator polynomial. The remainder of this division is the CRC, which is appended to the data and checked on receipt.

Different CRCs have different widths, commonly 8, 16, 32, or 64 bits. Each CRC is defined by a generator polynomial, an initial value, and a final XOR value, with optional bit-reflection. The choice of polynomial determines the kinds of errors the CRC is most likely to detect and affects implementation performance.

CRCs can be computed efficiently with linear feedback shift registers in hardware or with table-driven procedures in software. In many systems, the CRC is continuously updated as data are processed, and a nonzero remainder upon verification indicates corruption.

In terms of error-detection capabilities, CRCs are particularly good at detecting single-bit errors and burst errors up to the CRC width. For random errors, the undetected error probability is on the order of 2^{-r} for a CRC of width r. They do not provide cryptographic security and can miss some error patterns depending on the polynomial used.

CRC codes are widely used in network protocols, data storage, and file formats. Examples include CRC-32 in Ethernet and many software packages. They serve as a practical, lightweight means of detecting data integrity problems, rather than correcting errors.