Enumeration of Row-Column Designs
Gerold Jäger, Klas Markström, Lars-Daniel Öhman, Denys Shcherbak
TL;DR
Enumeration of row-column designs systematically catalogs binary, equireplicate $r\times c$ arrays on $v$ symbols, introducing mono arrays and AO-arrays alongside known sesqui, double, and triple arrays, and studying their autotopism groups. The authors deploy complete enumeration for $v\leq14$, develop constructive methods to realize all admissible AO-arrays, and use heuristic searches and SAT-based proofs to explore larger parameter sets, reporting extensive data and open questions. Key contributions include new design constructions, a detailed parameter-admissibility framework, connections to Youden rectangles and binary pseudo-Youden designs, and a comprehensive computational atlas of designs up to moderate sizes with explicit nonexistence results. The work advances both the theory and catalog of ordered row-column designs, with implications for experimental design and combinatorial structure, and highlights rich interactions with related design families. The findings offer practical constructions and pose precise open questions that guide future enumeration and construction efforts in this domain.
Abstract
We computationally completely enumerate a number of types of row-column designs up to isotopism, including double, sesqui and triple arrays as known from the literature, and two newly introduced types that we call mono arrays and AO-arrays. We calculate autotopism group sizes for the designs we generate. For larger parameter values, where complete enumeration is not feasible, we generate examples of some of the designs, and generate exhaustive lists of admissible parameters. For some admissible parameter sets, we prove non-existence results. We also give some explicit constructions of sesqui arrays, mono arrays and AO-arrays, and investigate connections to Youden rectangles and binary pseud Youden designs.