Multi-dimensional data structures

Abstract

In the examples so far the message block sizes have been determined by the size of the field used. If q is given by q = 2^m, where we are operating over GF(2^m), then each block has comprised q − 1, m bit symbols. There is, however, no reason why we can’t use smaller or larger message block sizes within the same field if we so choose. In the single-error correcting example of Chapter 6, (6.1) and (6.2) were used to force two roots into the message vector. We can construct the message with only three data symbols if we want, as follows:

$$ A + B + C + R + S = 0 and A\alpha ^1 + B\alpha ^2 + C\alpha ^3 + R\alpha ^6 + S\alpha ^7 = 0 $$

R and S are calculated in exactly the same way as before. We have actually increased the message redundancy by doing this and so we can expect a payback in terms of error information. We can still only correct one symbol but we now have a higher probability of detecting (not correcting) more than one error. Previously, if only one of the syndromes was non-zero we could infer that more than one error had occurred. The probability of this detection is now greater since if k, the error index, points to 4 or 5, we again realize that more than a single error has occurred. In this case we won’t attempt to correct the message. This is a shortened code.