Classification of base sequences BS(n+1,n)

TL;DR

This paper introduces a new equivalence definition and canonical form for base sequences BS(n+1,n), enumerates their classes for n up to 30, and discusses implications for the base sequence conjecture related to Hadamard matrices.

Contribution

It proposes a novel equivalence framework and provides enumeration results for base sequences, advancing understanding of their existence and structure.

Findings

Enumerated BS(n+1,n) for n ≤ 30.

Developed a canonical form for base sequences.

Provided partial tables for n ≤ 13.

Abstract

Base sequences BS(n+1,n) are quadruples of {1,-1}-sequences (A;B;C;D), with A and B of length n+1 and C and D of length n, such that the sum of their nonperiodic autocorrelation functions is a delta-function. The base sequence conjecture, asserting that BS(n+1,n) exist for all n, is stronger than the famous Hadamard matrix conjecture. We introduce a new definition of equivalence for base sequences BS(n+1,n) and construct a canonical form. By using this canonical form, we have enumerated the equivalence classes of BS(n+1,n) for n <= 30. Due to excessive size of the equivalence classes, the tables in the paper cover only the cases n <= 13.