The $q$-ary Golay complementary arrays of size $\mathbf{2}^{(m)}$ are standard

TL;DR

This paper proves that all $q$-ary Golay complementary array pairs of size $2^{(m)}$ are standard, confirming a key property of these arrays and advancing understanding of their structure.

Contribution

The paper demonstrates that every $q$-ary Golay complementary array pair of size $2^{(m)}$ is standard, resolving an open question in array theory.

Findings

All $q$-ary GCSs of length $2^m$ can be derived from standard $q$-ary Golay array pairs.

The nonexistence of non-standard binary Golay sequences of length $2^m$ is supported.

The result simplifies the classification of Golay arrays and sequences.

Abstract

To find the non-standard binary Golay complementary sequences (GCSs) of length $2^{m}$ or theoretically prove the nonexistence of them are still open. Since it has been shown that all the standard $q$-ary (where $q$ is even) GCSs of length $2^m$ can be obtained by standard $q$-ary Golay complementary array pair (GAP) of dimension $m$ and size $2\times 2 \times \cdots \times 2$ (abbreviated to size $\mathbf{2}^{(m)}$), it's natural to ask whether all the $q$-ary GAP of size $\mathbf{2}^{(m)}$ are standard. We give a positive answer to this question.