Near Triple Arrays

TL;DR

We introduce near triple arrays, a broad generalization of binary row-column designs that relax three intersection conditions to two adjacent values and allow two replication numbers, unifying triple arrays, (near) Youden rectangles, and Latin squares. The paper develops an orderly enumeration framework under isotopisms, achieves extensive computational results for numerous small parameter sets, and provides complete classifications for several key triples, accompanied by robust construction methods. A duality framework with near balanced grids links near triple arrays to MBMUDs, regular graph designs, PBDs, and BIBDs, yielding necessary conditions and nonexistence results via the duality theorems. Correctness checks via independent computations corroborate the results, and the work raises open questions and conjectures about the existence thresholds, generalized design parameters, and practical applications in experimental design efficiency.

Abstract

We introduce near triple arrays as binary row-column designs with at most two consecutive values for the replication numbers of symbols, for the intersection sizes of pairs of rows, pairs of columns and pairs of a row and a column. Near triple arrays form a common generalization of such well-studied classes of designs as triple arrays, (near) Youden rectangles and Latin squares. We enumerate near triple arrays for a range of small parameter sets and show that they exist in the vast majority of the cases considered. As a byproduct, we obtain the first complete enumerations of $6 \times 10$ triple arrays on $15$ symbols, $7 \times 8$ triple arrays on $14$ symbols and $5 \times 16$ triple arrays on $20$ symbols. Next, we give several constructions for families of near triple arrays, and e.g. show that near triple arrays with 3 rows and at least 6 columns exist for any number of symbols. Finally, we investigate a duality between row and column intersection sizes of a row-column design, and covering numbers for pairs of symbols by rows and columns. These duality results are used to obtain necessary conditions for the existence of near triple arrays. This duality also provides a new unified approach to earlier results on triple arrays and balanced grids.

Figures (10)

• Figure 1: A $4 \times 9$ triple array on $12$ symbols and a $4 \times 6$ near triple array on $9$ symbols. In the first, each symbol occurs 3 times, any two rows share 6 symbols, any two columns share 1 symbol, and any row and column share 3 symbols. In the second, each symbol occurs 2 or 3 times, any two rows share 3 or 4 symbols, any two columns share 1 or 2 symbols, and any row and column share 2 or 3 symbols.
• Figure 2: Near triple arrays on parameters (a) $(4 \times 6, 10)$, (b) $(4 \times 6, 7)$, (c) $(4 \times 7, 14)$ and (d) $(4 \times 6, 12)$. Designs (c) and (d) are equireplicate, and so have integer $\lambda_{rc}$. Designs (a), (b) and (d) have integer $\lambda_{rr}$, and design (b) has integer $\lambda_{cc}$.
• Figure 3: Examples of a $(7 \times 8, 28)$ and a $(8 \times 9, 36)$-near triple array.
• Figure 4: A $(4 \times 11, 13; 2,2,3)$-generalized triple array.
• Figure 5: Two $(4 \times 7, 9; 2,2,3)$-generalized triple arrays with $\lambda_{cc} = 1.42...$, with two columns sharing 0, 1 or 2 symbols in the first, and 1, 2 or 3 symbols in the second array, and a $(4 \times 6, 8; 2, 2, 3)$-generalized triple array with $\lambda_{cc} = 1.6$ and two columns sharing 0, 1, or 2 symbols.

Theorems & Definitions (55)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Remark 2.4
• proof : Proof of Lemma \ref{['lm:avg']}
• Definition 2.5
• Proposition 2.6
• proof