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

Jakub Czartowski

TL;DR

This work identifies and corrects an error in Baladram's claim of a 3-point simplex $3$-design in the $d$-point probability simplex by showing the cyclic symmetry-based argument is invalid. It introduces a corrected framework using the full symmetric group $\mathcal{S}_d$ to restore moment-equality conditions and further proposes symmetry-restricted simplex designs as a robust alternative. The authors extend the construction to a $d(d-1)$-point family that yields simplex $3$-designs for small dimensions and conjecture a general $d^{t-1}$ scaling for minimal designs, offering concrete equations and feasibility analyses. These contributions clarify the proper group-theoretic approach to simplex designs and open avenues for scalable, symmetry-aware design constructions with potential applications in numerical integration and related fields.

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.

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

TL;DR

This work identifies and corrects an error in Baladram's claim of a 3-point simplex -design in the -point probability simplex by showing the cyclic symmetry-based argument is invalid. It introduces a corrected framework using the full symmetric group to restore moment-equality conditions and further proposes symmetry-restricted simplex designs as a robust alternative. The authors extend the construction to a -point family that yields simplex -designs for small dimensions and conjecture a general scaling for minimal designs, offering concrete equations and feasibility analyses. These contributions clarify the proper group-theoretic approach to simplex designs and open avenues for scalable, symmetry-aware design constructions with potential applications in numerical integration and related fields.

Abstract

In [Bal18] a new method of constructing simplex designs based on cyclic group on elements has been proposed. One of the claims put forward therein is existence of 3-point simplex 3-design in dimension . 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 , which suggest a general scaling of . Finally, we put forward a notion that encompasses the objects conforming with bounds given in [Bal18], which we refer to as symmetry-restricted simplex -designs.
Paper Structure (12 sections, 5 theorems, 35 equations, 1 figure, 2 tables)

This paper contains 12 sections, 5 theorems, 35 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Comment on Lemma 3.1
  4. Evaluation of Corollary 4.2
  5. Decomposition of polynomials over probability simplex
  6. Correction for Theorem 3.2
  7. Discussion of correction
  8. Lemma 3.1 and Theorem 3.2 -- corrected versions
  9. Extension of results
  10. Extending proposed method of construction
  11. Symmetry restricted simplex designs
  12. Summary

Key Result

Lemma 1

Let $n, k\in\mathbb{Z}_{>0},\,r_1,\,r_2\in\mathbb{R}$. Define Let $V_k$ be a subspace spanned by $F(j,0,\hdots,0), 0\leq j\leq k$. Then, for non-negative integers $k_1,\hdots,k_n$ such that $\sum_{i=1}^n k_i = k$ one has $F(k_1,\hdots,k_n)\in V_k$.

Figures (1)

  • Figure 1: Monomials and symmetrised monomials in 3-point simplex: In the following figures we show contour plots of monomials $M(\vb{k})$ and symmetrised monomials $F_G(\vb{k})$ in 3-points simplex for $\sum k_i \leq 3$ and $G =\mathcal{C}_3,\,\mathcal{S}_3$; Additionally, we show cyclically symmetrised solution of \ref{['eq:3-point_non_des']} (red) and its mirror image (green). Note that for all $F_{\mathcal{S}_3}$ all six points lie on the same isoline of the function, thus noting that average over red and green points will be equal. However, in particular case of $F_{\mathcal{C}_3}(2,1,0)$, we find that red points lie on a different isoline than green points, thus leading to different values of averages depending on the mirror symmetry -- equivalent to statement given in Eq. \ref{['eq:av_inequal']}; Additionally, $F_{\mathcal{C}_3}(2,1,0)$ is the only polynomial without mirror symmetry, and thus cannot be decomposed of the remaining functions.

Theorems & Definitions (8)

  • Lemma 1: Lemma 3.1 from baladram2018explicit with $\mathcal{S}_d$
  • Theorem 1: Theorem 3.2 from baladram2018explicit with $\mathcal{S}_d$
  • proof
  • Theorem 2
  • Conjecture 1
  • Lemma 2: Lemma 3.1 from baladram2018explicit with $G$-invariant sequences $\vb{k}$
  • Theorem 3: Theorem 3.2 from baladram2018explicit with $G$-restricted designs
  • proof