The triplication method for constructing strong starters

TL;DR

This paper advances the triplication method for constructing strong starters in cyclic groups, generalizing the approach to include cases where the order is divisible by 3, thus broadening the scope of constructible strong starters.

Contribution

The authors generalize the triplication table and Sudoku-type problem formulation, enabling construction of strong starters for all odd orders, including those divisible by 3.

Findings

Expanded triplication table definition includes pseudostarters.

Broadened Sudoku-type problem formulation for various modular encodings.

Eliminated the restriction that the order must not be divisible by 3.

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.