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Combinatorial characterzations of $T$-designs in the nonbinary Johnson scheme

Hiroshi Nozaki, Yuta Watanabe

TL;DR

This work extends Delsarte-style characterizations of T-designs to the nonbinary Johnson scheme, introducing the (r,s)-designs as a unifying framework that encompasses block designs and orthogonal arrays. It provides a purely combinatorial criterion for T-designs in J_q(w,n) and derives Fisher-type lower bounds on the size of such designs via the scheme's eigenstructure, supported by explicit constructions of minimal (λ=1) designs. The results unify classical designs within a single framework and open avenues for multivariate generalizations in multivariate Q-polynomial schemes. The paper also discusses potential nonexistence results and future directions for linear-programming-type bounds in this multivariate setting.

Abstract

We study $T$-designs in the nonbinary Johnson scheme. This scheme generalizes both the Johnson and Hamming schemes and admits a bivariate $Q$-polynomial structure. Zhu (2021) provided a combinatorial characterization of $T$-designs in this scheme for certain index sets $T$, using a relationship between $T$-designs in the nonbinary Johnson scheme and relative designs in the nonbinary Hamming scheme. In this paper, we obtain a characterization that applies to a strictly larger class of index sets $T$, based on a methodological extension of Delsarte's original framework (1973). This new characterization naturally recovers classical block designs and orthogonal arrays as special cases. To describe these designs uniformly, we introduce $(r,s)$-designs, a new family of combinatorial objects that arise naturally from our characterization. We also derive absolute lower bounds on the cardinality of $(r,s)$-designs from the multiplicities of the primitive idempotents of the nonbinary Johnson scheme, and construct examples with index $λ=1$ that attain certain natural lower bounds.

Combinatorial characterzations of $T$-designs in the nonbinary Johnson scheme

TL;DR

This work extends Delsarte-style characterizations of T-designs to the nonbinary Johnson scheme, introducing the (r,s)-designs as a unifying framework that encompasses block designs and orthogonal arrays. It provides a purely combinatorial criterion for T-designs in J_q(w,n) and derives Fisher-type lower bounds on the size of such designs via the scheme's eigenstructure, supported by explicit constructions of minimal (λ=1) designs. The results unify classical designs within a single framework and open avenues for multivariate generalizations in multivariate Q-polynomial schemes. The paper also discusses potential nonexistence results and future directions for linear-programming-type bounds in this multivariate setting.

Abstract

We study -designs in the nonbinary Johnson scheme. This scheme generalizes both the Johnson and Hamming schemes and admits a bivariate -polynomial structure. Zhu (2021) provided a combinatorial characterization of -designs in this scheme for certain index sets , using a relationship between -designs in the nonbinary Johnson scheme and relative designs in the nonbinary Hamming scheme. In this paper, we obtain a characterization that applies to a strictly larger class of index sets , based on a methodological extension of Delsarte's original framework (1973). This new characterization naturally recovers classical block designs and orthogonal arrays as special cases. To describe these designs uniformly, we introduce -designs, a new family of combinatorial objects that arise naturally from our characterization. We also derive absolute lower bounds on the cardinality of -designs from the multiplicities of the primitive idempotents of the nonbinary Johnson scheme, and construct examples with index that attain certain natural lower bounds.
Paper Structure (7 sections, 10 theorems, 62 equations, 2 figures)

This paper contains 7 sections, 10 theorems, 62 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Combinatorial characterizations of $T$-designs in $J_q(w,n)$
  4. Combinatorial properties of $(r,s)$-designs
  5. Lower bounds on $(r,s)$-designs
  6. Minimal $(r,s)$-designs with $\lambda=1$
  7. Concluding remarks

Key Result

Lemma 3.1

Let $(r,s) \in L$. The set of components of $C_{rs} \in \mathfrak{B}$ is

Figures (2)

  • Figure 1: $(2,1)$-$(5,3,4,1)$-design with $10$ vectors
  • Figure 2: $(2,1)$-$(6,4,3,3)$-design with $15$ vectors

Theorems & Definitions (26)

  • Definition 1.1: $(r,s)$-design
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Definition 4.1: $(r,s)$-design
  • Proposition 4.2
  • proof
  • Proposition 4.3
  • proof
  • ...and 16 more