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<p>Generalized Howell designs (GHDs) are doubly resolvable designs which can be displayed in a square array. They are generalizations of Howell designs and Kirkman squares. In this paper, we investigate the existence of generalized Howell designs with block size four. Using starters and adders together with Weil’s theorem, we establish asymptotic existence results for GHDs of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(3p + 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mi>p</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> for prime <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(p\equiv 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (mod 4), except when <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p\equiv 1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≡</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> (mod 32). In addition, several classes of optimal multiply constant-weight codes with weight 6 and distance 10 are obtained as applications of these designs.</p>

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On generalized Howell designs with block size four and their applications to multiply constant-weight codes

  • 期刊论文

  • 发表日期：

    2026年4月7日

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  • Xian Zhao,
  • Zihong Tian,
  • Guohui Hao

摘要

Generalized Howell designs (GHDs) are doubly resolvable designs which can be displayed in a square array. They are generalizations of Howell designs and Kirkman squares. In this paper, we investigate the existence of generalized Howell designs with block size four. Using starters and adders together with Weil’s theorem, we establish asymptotic existence results for GHDs of order \(3p + 1\) 3 p + 1 for prime \(p\equiv 1\) p ≡ 1 (mod 4), except when \(p\equiv 1\) p ≡ 1 (mod 32). In addition, several classes of optimal multiply constant-weight codes with weight 6 and distance 10 are obtained as applications of these designs.