Error Correction Using the CRC

Abstract

Hamming codes work well for small messages where parallel implementation of the computation is possible, but for longer messages the CRC provides a more economic solution. A single-bit error can be corrected provided the CRC is long enough to describe every bit location within the protected part of the message, and that the CRC is based on a primitive generator polynomial. If the CRC is n bits, then the total protected message length (i.e. including the CRC itself) must be less than 2ⁿ bits. A CRC encoded message, d ₀ to d _{m −1}, (m = 2ⁿ − 1) satisfies the equation

$$ \sum\limits_{i = 0}^{m - 1} {d_i \alpha ^i = 0.} $$

.