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<p>Geometric orthogonal codes (GOCs) were introduced by Doty and Winslow to address problems related to DNA origami technology. Wang et al. introduced the concepts of geometric difference packing (GDP) and geometric difference family (GDF) and established the equivalence between GOCs and GDPs. In this paper, we present some recursive constructions for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\((n\times m, k, \lambda , 1)\)</EquationSource> </InlineEquation>-GDFs by using Langford sequences and perfect difference matrices as auxiliary designs. As a consequence, the perfect <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\((n\times m,3,2,1)\)</EquationSource> </InlineEquation>-GOCs are given for any positive integers <i>n</i> and <i>m</i>.</p>

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The existence of perfect \((n\times m, 3, 2, 1)\)-geometric orthogonal codes

  • 期刊论文

  • 发表日期：

    2026年2月3日

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  • Huizhen Gao,
  • Zihong Tian

摘要

Geometric orthogonal codes (GOCs) were introduced by Doty and Winslow to address problems related to DNA origami technology. Wang et al. introduced the concepts of geometric difference packing (GDP) and geometric difference family (GDF) and established the equivalence between GOCs and GDPs. In this paper, we present some recursive constructions for \((n\times m, k, \lambda , 1)\) -GDFs by using Langford sequences and perfect difference matrices as auxiliary designs. As a consequence, the perfect \((n\times m,3,2,1)\) -GOCs are given for any positive integers n and m.