Comment and correction for "On Explicit Construction of Simplex $t$-designs" by M. S. Baladram

TL;DR

This paper critiques and clarifies previous work on simplex t-designs, correcting proofs and proposing a new framework called symmetry-restricted simplex t-designs, with implications for their construction and scaling.

Contribution

It provides explicit counterarguments to prior claims, rectifies proofs, and introduces the concept of symmetry-restricted simplex t-designs for better understanding and construction.

Findings

Counterarguments to previous claims about 3-point simplex 3-designs.

A method to construct simplex 3-designs scaling as d^2.

Introduction of symmetry-restricted simplex t-designs.

Abstract

In [Bal18] a new method of constructing simplex designs based on cyclic group on $n$ elements has been proposed. One of the claims put forward therein is existence of 3-point simplex 3-design in dimension $d = 3$. In this manuscript we present explicit counterarguments and suggest a manner to rectify the existing proofs. By doing this, we show that the results presented in [Bal18] can be utilised to construct simplex 3-designs scaling as $d^2$, which suggest a general scaling of $d^{t-1}$. Finally, we put forward a notion that encompasses the objects conforming with bounds given in [Bal18], which we refer to as symmetry-restricted simplex $t$-designs.