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 m bits, then the total protected message length (i.e. including the CRC itself) must be less than 2^m bits. The generator polynomial used in Chapter 3 was x⁴ + x³ + 1. Using the primitive element α = 2 (or x), a finite field can be constructed in the same way as shown in Table 3.1, except in this case there will be 15 nonzero elements rather than seven owing to the larger (five-bit) primitive polynomial. This time the elements are listed in Table 5.1.