<p>An <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(m,n,k,\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-relative difference set is a lifting of an <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(m,k,n\lambda )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-difference set. Lam gave a table of cyclic relative difference sets with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$k \le 50$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≤</mo> <mn>50</mn> </mrow> </math></EquationSource> </InlineEquation> in 1977, all of which were liftings of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$( \frac{q^d-1}{q-1},q^{d-1},q^{d-2}(q-1))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mfrac> <mrow> <msup> <mi>q</mi> <mi>d</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> <mrow> <mi>q</mi> <mo>-</mo> <mn>1</mn> </mrow> </mfrac> <mo>,</mo> <msup> <mi>q</mi> <mrow> <mi>d</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow> <mi>d</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-difference sets, the parameters of complements of classical Singer difference sets. Pott found all liftings of these difference sets with <i>n</i> odd and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$k \le 64$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≤</mo> <mn>64</mn> </mrow> </math></EquationSource> </InlineEquation> in 1995. No other nontrivial difference sets are known with liftings to relative difference sets, and Pott ended his survey on relative difference sets asking whether there are any others. In this paper, we extend these searches and apply the results to the existence of circulant weighing matrices.</p>

Cyclic relative difference sets and circulant weighing matrices

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发表日期：

2026年2月28日

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Daniel M. Gordon

摘要

An $(m,n,k,\lambda )$ $(m, n, k, λ)$ -relative difference set is a lifting of an $(m,k,n\lambda )$ $(m, k, n λ)$ -difference set. Lam gave a table of cyclic relative difference sets with $k \le 50$ $k \leq 50$ in 1977, all of which were liftings of $( \frac{q^d-1}{q-1},q^{d-1},q^{d-2}(q-1))$ $(\frac{q^{d} - 1}{q - 1}, q^{d - 1}, q^{d - 2} (q - 1))$ -difference sets, the parameters of complements of classical Singer difference sets. Pott found all liftings of these difference sets with n odd and $k \le 64$ $k \leq 64$ in 1995. No other nontrivial difference sets are known with liftings to relative difference sets, and Pott ended his survey on relative difference sets asking whether there are any others. In this paper, we extend these searches and apply the results to the existence of circulant weighing matrices.