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<p>Covering schemes were recently introduced as a generalization of both difference matrices and difference schemes, providing an efficient method for building covering arrays with a certain degree of symmetry. On the other hand, Tang and Woo (1983) showed that sets of constant-weight tuples yield several upper bounds on covering arrays. However, these sets usually do not have enough symmetry to induce good covering schemes. In this work, we derive a classification of invariant sets of constant-weight tuples under the action of the diagonal group of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {Z}_q^k\)</EquationSource> </InlineEquation>. This classification enables us to obtain new upper bounds on covering scheme numbers, including the exact class for binary covering scheme numbers with <i>k</i> columns and strength <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(k-2.\)</EquationSource> </InlineEquation></p>

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A classification of invariant sets of constant-weight tuples and applications to covering schemes

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年4月27日

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  • André G. Castoldi,
  • Anderson N. Martinhão,
  • Emerson L. Monte Carmelo,
  • Pablo H. Perondi,
  • Otávio J. N. T. N. dos Santos

摘要

Covering schemes were recently introduced as a generalization of both difference matrices and difference schemes, providing an efficient method for building covering arrays with a certain degree of symmetry. On the other hand, Tang and Woo (1983) showed that sets of constant-weight tuples yield several upper bounds on covering arrays. However, these sets usually do not have enough symmetry to induce good covering schemes. In this work, we derive a classification of invariant sets of constant-weight tuples under the action of the diagonal group of \(\mathbb {Z}_q^k\) . This classification enables us to obtain new upper bounds on covering scheme numbers, including the exact class for binary covering scheme numbers with k columns and strength \(k-2.\)