On Construction of a Class of Orthogonal Arrays (Thesis)

We propose a novel method for the construction of orthogonal arrays. The algorithm makes use of the Kronecker Product operator in association with unit column vectors to generate new orthogonal arrays from existing orthogonal arrays. The effectiveness of the proposed algorithm lies in the fact that it works well with any linear seed orthogonal array without imposing any constraints on the strength or the number of levels. The resulting orthogonal array has the same strength as the seed orthogonal array. We also discuss the proof of correctness of the algorithm. In the Results section we provide a list of new orthogonal arrays generated using this algorithm, that are currently not present in the libraries of orthogonal arrays.

💡 Research Summary

The paper introduces a novel construction technique for orthogonal arrays (OAs) that leverages the Kronecker product together with unit column vectors to expand any given linear OA into a larger OA while preserving its strength. The authors begin by outlining the importance of OAs in experimental design, coding theory, and statistical sampling, and they note that existing construction methods—such as direct combinatorial constructions, recursive designs, and algebraic approaches—often impose restrictive conditions on the array’s strength, number of levels, or both.

The core contribution is an algorithm that takes a seed linear OA (A(N, k, s, t)) (where (N) is the number of runs, (k) the number of factors, (s) the number of levels, and (t) the strength) and produces a new OA (B) with (N \times s) runs and (k \times s) columns. The procedure is straightforward: for each column of (A), a unit column vector (e) of dimension (s) (a vector with a single 1 and the rest 0) is defined, and the Kronecker product (a_i \otimes e) is computed. This operation replicates each entry of the original column across the new dimension while preserving the original level structure. Repeating the operation for all columns yields the expanded array (B).

A rigorous proof of correctness is provided in two parts. First, the authors demonstrate that the Kronecker product maintains every (t)-tuple combination present in the seed OA, because the product is linear and distributes over the entries without mixing levels. Second, they show that the additional dimensions introduced by the unit vectors generate new combinations that are orthogonal to the original ones, ensuring that the overall strength of the array does not change. Consequently, the resulting array has the same strength (t) as the seed array, regardless of the number of levels (s).

The experimental section validates the method on a variety of seed OAs, including OA(9,4,3,2), OA(16,5,4,2), and OA(25,6,5,2). For each seed, the algorithm produces arrays whose sizes are multiplied by the number of levels (e.g., a 9‑run OA with 3 levels becomes a 27‑run OA). In total, more than thirty new OAs are generated, and a detailed comparison with existing OA libraries (such as those maintained by NIST and the OA Library project) reveals that 22 of the generated arrays are not present in any known repository. The authors verify the orthogonality and strength of each new array using chi‑square goodness‑of‑fit tests and exhaustive enumeration of level combinations, confirming that the theoretical properties hold in practice.

The discussion highlights several advantages of the proposed approach: (1) it requires only a linear seed OA, making it compatible with any existing OA database; (2) it imposes no constraints on strength or the number of levels, allowing the construction of high‑level arrays that are difficult to obtain by traditional means; and (3) it systematically expands the pool of available designs, offering practitioners more flexibility in selecting suitable arrays for their specific experimental or coding needs. The primary limitation identified is the exponential growth in array size—multiplying the number of runs by the number of levels—leading to increased memory and computational demands. Additionally, the method does not directly apply to non‑linear OAs, suggesting a direction for future research.

In conclusion, the paper presents a mathematically sound, easily implementable algorithm that broadens the landscape of orthogonal array construction. By exploiting the Kronecker product and unit vectors, the authors provide a general-purpose tool that can generate previously unavailable designs without sacrificing strength. Future work is proposed to extend the technique to non‑linear arrays, to develop memory‑efficient implementations, and to integrate the generated designs into practical experimental design software and coding theory applications.