14 - Coding systems for error control

Abstract

This chapter presents coding systems for error control. A code consists of a specified number of binary digits and if the code is uniform, each codeword consists of the same number of binary digits k and the corresponding number of code combinations is 2k. If all of these combinations are used, the code is complete and irredundant. Each codeword occupies one cell on the map and codewords in adjacent cells differ in one digit place only; hence, the minimum distance between adjacent codewords is dmin = 1. If a single-bit error occurs in any of the tabulated codewords, an alternative valid codeword is generated. The code map shows that for a correction domain having a square of sides distance d, it is possible to detect (d - 1) bit errors and correct (d - 1)/2 bit errors. For d = 5, up to and including 4-bit errors can be detected and up to and including 2-bit errors can be corrected. If the probability of a single-bit error occurring in the transmission channel is p, then the merit of a coding scheme is defined by p', the undetected error rate. A criterion for a good coding scheme is that the undetected error rate is very much less than the bit error rate, that p' ≪ p. To determine the undetected error rate p', the probability of double and quadruple compensating errors has first to be obtained.