The triplication method for constructing strong starters

Abstract

The triplication method for constructing strong starters in $Z_{3m}$ from starters in $Z_{m}$ (say, a starter of order 21 from a starter of order 7) was proposed by the authors in 2025. The method reduced construction of the particular combinatorial design (a strong starter in a cyclic group) to solving a Sudoku-type problem -- an independent task with its own tools and techniques available. The Sudoku-type problem was formulated in terms of the so-called triplication table constructed from a starter of order $m$. The method was applicable for odd orders $m\ge 7$ not divisible by 3. In the present paper, our previous approach is developed in two directions: (1) the definition of the triplication table is generalized, which expands possibilities for its construction to include three base starters or even ``pseudostarters''; (2) the formulation of the Sudoku-type problem is broadened to embrace various scenarios of ``modular encoding'' and reconstruction of strong starters from its solution. A theoretical gain of these developments consists in the improved understanding of the general structure of the triplication approach. A practical outcome is elimination of the requirement that $m$ be not divisible by 3. This leads to a broader scope of strong starters obtainable by triplication: any latent strong starter of odd order $3m$ can emerge this way.