On the base sequence conjecture

TL;DR

This paper advances the base sequence conjecture by proving its validity for n=37 and n=38, extending computational results of associated graphs, and proposing a general conjecture about their structure.

Contribution

It proves the base sequence conjecture for n=37 and 38 and extends the computational analysis of related graphs Gamma_n for n=28 to 35.

Findings

Confirmed existence of base sequences for n=37 and 38.

Extended the computation of Gamma_n graphs up to n=35.

Proposed a conjecture describing the structure of Gamma_n graphs.

Abstract

Let BS(m,n) denote the set of base sequences (A;B;C;D), with A and B of length m and C and D of length n. The base sequence conjecture (BSC) asserts that BS(n+1,n) exist (i.e., are non-empty) for all n. This is known to be true for n <= 36 and when n is a Golay number. We show that it is also true for n=37 and n=38. It is worth pointing out that BSC is stronger than the famous Hadamard matrix conjecture. In order to demonstrate the abundance of base sequences, we have previously attached to BS(n+1,n) a graph Gamma_n and computed the Gamma_n for n <= 27. We now extend these computations and determine the Gamma_n for n=28,...,35. We also propose a conjecture describing these graphs in general.