Turyn-type sequences: Classification, Enumeration and Construction

TL;DR

This paper introduces a canonical form for Turyn-type sequences, classifies and enumerates them up to length 32, and constructs the first sequence of length 38, advancing combinatorial sequence theory.

Contribution

It defines a new equivalence and canonical form for TT(n), enabling enumeration and the first construction of TT(38).

Findings

Enumerated TT(n) for n up to 32.

Constructed the first TT(38).

Established a new classification method.

Abstract

Turyn-type sequences, TT(n), are quadruples of {+,-1}-sequences (A;B;C;D), with lengths n,n,n,n-1 respectively, where the sum of the nonperiodic autocorrelation functions of A,B and twice that of C,D is a delta-function (i.e., vanishes everywhere except at 0). Turyn-type sequences TT(n) are known to exist for all even n not larger than 36. We introduce a definition of equivalence to construct a canonical form for TT(n) in general. By using this canonical form, we enumerate the equivalence classes of TT(n) for n up to and including 32. We also construct the first example of Turyn-type sequences TT(38).