Let q be a prime and let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\lambda >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> be an integer coprime to q such that <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> is self-conjugate modulo <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda q$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mi>q</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\gcd (\lambda ,q-1)=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo movablelimits="true">gcd</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>,</mo> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> or 2. Suppose a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$(\lambda q,q,\lambda q,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>λ</mi> <mi>q</mi> <mo>,</mo> <mi>q</mi> <mo>,</mo> <mi>λ</mi> <mi>q</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> relative difference set D exists in an abelian group G. Then <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> is a square and D admits a (q, q, q, 1) relative difference set as a sub-difference set. Moreover, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$q=3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>q</mi> <mo>=</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and the Sylow q-subgroup of G is isomorphic to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$C_3\times C_3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>3</mn> </msub> <mo>×</mo> <msub> <mi>C</mi> <mn>3</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>. If p is an odd prime dividing <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$p^{4b}||\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>p</mi> <mrow> <mn>4</mn> <mi>b</mi> </mrow> </msup> <mrow> <mo stretchy="false">|</mo> <mo stretchy="false">|</mo> </mrow> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> for some positive integer b.

Semiregular relative difference sets related to Gauss sums and projective planes

期刊论文
发表日期：

2025年9月27日

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Ka Hin Leung,
Bernhard Schmidt,
Tao Zhang

摘要

Let q be a prime and let $\lambda >1$ $λ > 1$ be an integer coprime to q such that $\lambda $ $λ$ is self-conjugate modulo $\lambda q$ $λ q$ and $\gcd (\lambda ,q-1)=1$ $gcd (λ, q - 1) = 1$ or 2. Suppose a $(\lambda q,q,\lambda q,\lambda )$ $(λ q, q, λ q, λ)$ relative difference set D exists in an abelian group G. Then $\lambda $ $λ$ is a square and D admits a (q, q, q, 1) relative difference set as a sub-difference set. Moreover, $q=3$ $q = 3$ and the Sylow q-subgroup of G is isomorphic to $C_3\times C_3$ $C_{3} \times C_{3}$ . If p is an odd prime dividing $\lambda $ $λ$ , then $p^{4b}||\lambda $ $p^{4 b} | | λ$ for some positive integer b.