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<p>Special polynomials have interesting applications in combinatorial designs. Recently, the image sets of a fixed size of certain special polynomials were used to construct a <i>t</i>-design. However, determining the parameters in these designs is not an easy task in general. In 2020, Xiang et al. (Des. Codes Cryptogr. 88:553–565, 2020) determined the parameters of 2-designs from the Gold function in some cases. In this paper, we use the Gold function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(x^{2^h+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>x</mi> <mrow> <msup> <mn>2</mn> <mi>h</mi> </msup> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> over <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {F}_{2^e}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mn>2</mn> <mi>e</mi> </msup> </msub> </math></EquationSource> </InlineEquation> to construct 2-designs and determine their parameters by some Weil sums. Our method to determine the size of the stabiliser of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(B_h\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mi>h</mi> </msub> </math></EquationSource> </InlineEquation> is completely different from the one used in Xiang et al. (Des. Codes Cryptogr. 88:553–565, 2020). Moreover, we explicitly determine the parameters of 2-designs from a class of <i>d</i>-monomials and confirm Conjecture 1 proposed by Xiang et al. (Des. Codes Cryptogr. 88:553–565, 2020).</p>

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Constructions of t-designs from the gold function

  • 期刊论文

  • 发表日期：

    2026年5月16日

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  • Guangkui Xu,
  • Xiwang Cao,
  • Gaojun Luo

摘要

Special polynomials have interesting applications in combinatorial designs. Recently, the image sets of a fixed size of certain special polynomials were used to construct a t-design. However, determining the parameters in these designs is not an easy task in general. In 2020, Xiang et al. (Des. Codes Cryptogr. 88:553–565, 2020) determined the parameters of 2-designs from the Gold function in some cases. In this paper, we use the Gold function \(x^{2^h+1}\) x 2 h + 1 over \(\mathbb {F}_{2^e}\) F 2 e to construct 2-designs and determine their parameters by some Weil sums. Our method to determine the size of the stabiliser of \(B_h\) B h is completely different from the one used in Xiang et al. (Des. Codes Cryptogr. 88:553–565, 2020). Moreover, we explicitly determine the parameters of 2-designs from a class of d-monomials and confirm Conjecture 1 proposed by Xiang et al. (Des. Codes Cryptogr. 88:553–565, 2020).