Low Autocorrelation Binary Sequences: Number Theory-based Analysis for Minimum Energy Level, Barker codes

TL;DR

This paper uses number theory to analyze low autocorrelation binary sequences, deriving theoretical minimum energy levels, explaining the existence of Barker sequences up to length 13, and supporting the conjecture that Barker sequences do not exist beyond that length.

Contribution

The paper proves the finiteness of possible energy levels, derives the theoretical minimum energy levels for sequences of even and odd lengths, and explains the existence of Barker sequences up to length 13.

Findings

Finite energy levels are spaced at intervals of 4.

Theoretical minimum energy levels are N/2 for even N and (N-1)/2 for odd N.

Barker sequences exist only for sequence lengths up to 13.

Abstract

Low autocorrelation binary sequences (LABS) are very important for communication applications. And it is a notoriously difficult computational problem to find binary sequences with low aperiodic autocorrelations. The problem can also be stated in terms of finding binary sequences with minimum energy levels or maximum merit factor defined by M.J.E. Golay, F=N^2/2E, N and E being the sequence length and energy respectively. Conjectured asymptotic value of F is 12.32 for very long sequences. In this paper, a theorem has been proved to show that there are finite number of possible energy levels, spaced at an equal interval of 4, for the binary sequence of a particular length. Two more theorems are proved to derive the theoretical minimum energy level of a binary sequence of even and odd length of N to be N/2, and N-1/2 respectively, making the merit factor equal to N and N^2/N-1 respectively. The derived theoretical minimum energy level successfully explains the case of N =13, for which the merit factor (F =14.083) is higher than the conjectured value. Sequence of lengths 4, 5, 7, 11, 13 are also found to be following the theoretical minimum energy level. These sequences are exactly the Barker sequences which are widely used in direct-sequence spread spectrum and pulse compression radar systems because of their low autocorrelation properties. Further analysis shows physical reasoning in support of the conjecture that Barker sequences exists only when N <= 13 (this has been proven for all odd N).