Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1570_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="78" /> </InlineMediaObject> <EquationSource Format="TEX">$n=2^k-1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <msup> <mn>2</mn> <mi>k</mi> </msup> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1570_Article_IEq2.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="68" /> </InlineMediaObject> <EquationSource Format="TEX">$m=2^{k-2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>=</mo> <msup> <mn>2</mn> <mrow> <mi>k</mi> <mo>-</mo> <mn>2</mn> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> for a certain <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1570_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>. Consider the point-line geometry of 2m-element subsets of an n-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension k. For <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1570_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k\ge 4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric (n, 2m, m)-designs. The main results concern the case <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1570_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$k=4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> and give a geometric interpretation of the five well-known symmetric (15, 8, 4)-designs.

Symmetric (15, 8, 4)-designs in terms of the geometry of binary simplex codes of dimension 4

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发表日期：

2025年1月24日

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Mark Pankov,
Krzysztof Petelczyc,
Mariusz Żynel

摘要

Let $n=2^k-1$ $n = 2^{k} - 1$ and $m=2^{k-2}$ $m = 2^{k - 2}$ for a certain $k\ge 3$ $k \geq 3$ . Consider the point-line geometry of 2m-element subsets of an n-element set. Maximal singular subspaces of this geometry correspond to binary simplex codes of dimension k. For $k\ge 4$ $k \geq 4$ the associated collinearity graph contains maximal cliques different from maximal singular subspaces. We investigate maximal cliques corresponding to symmetric (n, 2m, m)-designs. The main results concern the case $k=4$ $k = 4$ and give a geometric interpretation of the five well-known symmetric (15, 8, 4)-designs.