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The triplication method for constructing strong starters

Oleg Ogandzhanyants, Sergey Sadov, Margo Kondratieva

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.

The triplication method for constructing strong starters

Abstract

The triplication method for constructing strong starters in from starters in (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 . The method was applicable for odd orders 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 be not divisible by 3. This leads to a broader scope of strong starters obtainable by triplication: any latent strong starter of odd order can emerge this way.
Paper Structure (24 sections, 16 theorems, 45 equations, 2 figures, 11 tables)

This paper contains 24 sections, 16 theorems, 45 equations, 2 figures, 11 tables.

Table of Contents

  1. Introduction
  2. The triplication table
  3. Definition of the triplication table $\Sigma_m$
  4. Objects associated with a triplication table $\Sigma_m$
  5. The Modular Sudoku Problem
  6. Discriminating 3-to-1 modular reduction $\mathbb{Z}_{3m}\to \mathbb{Z}_m$
  7. "Arithmetic" on discriminators
  8. A table $\tilde{\Sigma}^S$ consisting of discriminating values
  9. Congruous table, Modular Sudoku Problem, and the recovery of a strong starter
  10. Scenario-independence of the set of congruous starters
  11. Explicit constructions of the triplication table
  12. Triplication template, admissible keys
  13. Triplication based on one starter of order $m$
  14. Triplication based on three starters of order $m$
  15. Triplication with epicycloidal pseudostarters
  16. ...and 9 more sections

Key Result

Theorem 2.1

Let $S$ be a strong starter in $\mathbb{Z}_{3m}$, $m=2q+1$, $q\geq 1$. Let $S_m=\{\{x\!\!\mod m,\, y\!\!\mod m\}\mid \{x,y\}\in S\}$. Then (i) for $x\in\mathbb{Z}_m$, the multiplicity of $x$ in $\uplus{S_m}$ equals $3$ if $x\neq 0$ and $2$ if $x=0$; (ii) $S_m$ contains exactly one pair of type $\{t,

Figures (2)

  • Figure 1: Schemes of the triplication method: Mod scenario (left) and Carry scenario (right). Top row: from strong starter to tables (derivation of conditions and constraints). Bottom row: from triplication table $\Sigma_m$ to strong starters.
  • Figure 2: A discrimination scenario

Theorems & Definitions (58)

  • Theorem 2.1
  • proof
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • ...and 48 more