Generating Functions Associated with Random Binary Sequences Consisting of Runs of Lengths 1 and 2

Abstract

We associate a generating function of two formal variables with a given binary sequence and a generating function of three formal variables with a given pair of binary sequences. The first function gives information about all subsequences of the sequence and the second function gives information about all common subsequences of the pair of sequences. It is shown that, in many cases, these functions can be easily found, which is of interest for various applications such as reconstruction of sequences, pattern recognition, data transmission over channels with deletions, etc. [1],[2],[3]. This conclusion is demonstrated for random sequences chosen from a completely randomized probabilistic ensemble of binary sequences and from the ensemble of random sequences consisting of runs of lengths 1 and 2. The results show that the latter ensemble can be considered as a very good candidate for the ensemble of random codes capable of correcting deletion errors.

This work was supported by the Philips Research Laboratories (The Netherlands) and DFG (Germany).