Linear complexity of transformed sequences

Abstract

This paper deals with the effect of bit change errors on the linear complexity of finite sequences. Though a change in a single bit can cause a large change in linear complexity, it is shown that on the average the change will be small even when many bits, e.g. 10%, are changed. General bijections on the set of sequences of length n are studied and tight bounds are found on the average difference in linear complexity between a sequence and its image. It is also shown that to change all sequences of length n into sequences with linear complexity less than c(n) where limn→∞ c(n)/n = 0, at least n-1/n 2n of the sequences must have close to half of their bits changed.