Hadley, Lucinda and Briggs, Keith (2016) The Generation of Badly Approximable Pairs for Communications via Real Interference Alignment. In: 11th IMA International Conference on Mathematics in Signal Processing, 2016-12-12 - 2018-08-14, IET Austin Court.
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Abstract
Certain proposed coding schemes require sets of irrational numbers (a1,a2,...,an) which generate linear forms. It is conjectured that the more badly approximable these linear forms, meaning the greater the positive lower bound of qn|q+p1a1+...+pnan| for any choice of integers (q,p1, ..., pn), the better the coding scheme, thus the lower the error rate. In contrast to classical one-dimensional Diophantine approximation theory (n=1), the situation for n>1 is full of unsolved problems, and it is not even known what the worst approximable pair is. The aim of this paper will be to present some purely numerical results which suggest some good candidates for bad pairs, and to demonstrate the performance of these pairs in a transmission protocol. For this we use an algorithm due to Vaughan Clarkson, but the software implementation requires some very delicate treatment of floating-point arithmetic. This results in the first fully-rigorous implementation of an algorithm for finding the sequence of best approximants for a linear form q+p1a1+p2a2, and for the simultaneous rational approximation of two irrationals, and we demonstrate the effect of using such linear forms on the error rate of our coding scheme.