On the Smith normal form of a skew-symmetric D-optimal design of order $n\equiv 2\pmod{4}$

TL;DR

This paper determines the Smith normal form of skew-symmetric D-optimal designs of order n ≡ 2 mod 4, explicitly expressing it in terms of n, and uses this to analyze design equivalences and correct previous literature.

Contribution

It explicitly characterizes the Smith normal form for these designs based on their order, confirming a recent conjecture and correcting earlier results.

Findings

Smith normal form is determined by the order n.

Explicit formula for the Smith normal form in terms of n.

Certain designs are not equivalent to any skew-symmetric D-optimal design.

Abstract

We show that the Smith normal form of a skew-symmetric D-optimal design of order $n\equiv 2\pmod{4}$ is determined by its order. Furthermore, we show that the Smith normal form of such a design can be written explicitly in terms of the order $n$, thereby proving a recent conjecture of Armario. We apply our result to show that certain D-optimal designs of order $n\equiv 2\pmod{4}$ are not equivalent to any skew-symmetric D-optimal design. We also provide a correction to a result in the literature on the Smith normal form of D-optimal designs.