The weight spectra of the Reed–Muller codes RM(r, m) were unknown for <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1708_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="112" /> </InlineMediaObject> <EquationSource Format="TEX">$r=3,...,m-5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>=</mo> <mn>3</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>m</mi> <mo>-</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. In IEEE Trans. Inform. Theory 2024, Carlet determined the weight spectrum of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1708_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$RM(m-5,m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>-</mo> <mn>5</mn> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1708_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$m\ge 10$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>10</mn> </mrow> </math></EquationSource> </InlineEquation> using the Maiorana-McFarland construction, where he also studied the difficulties of trying to extend the results to <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1708_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$RM(m-6,m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>-</mo> <mn>6</mn> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we fully determine the weight spectrum of <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1708_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$RM(m-6,m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>-</mo> <mn>6</mn> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1708_Article_IEq8.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$m\ge 12$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>≥</mo> <mn>12</mn> </mrow> </math></EquationSource> </InlineEquation>, which gives a positive answer to an open question on the weight spectrum of <InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1708_Article_IEq9.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="110" /> </InlineMediaObject> <EquationSource Format="TEX">$RM(m-c,m)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>M</mi> <mo stretchy="false">(</mo> <mi>m</mi> <mo>-</mo> <mi>c</mi> <mo>,</mo> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10623_2025_1708_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$c=6$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>c</mi> <mo>=</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we put forward a conjecture and verify it for some cases. If the conjecture is true, that open question can be completely solved.

Determining the weight spectrum of the Reed–Muller codes $RM(m-6,m)$

期刊论文
发表日期：

2025年8月8日

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Yueying Lou,
Qichun Wang

摘要

The weight spectra of the Reed–Muller codes RM(r, m) were unknown for $r=3,...,m-5$ $r = 3, . . ., m - 5$ . In IEEE Trans. Inform. Theory 2024, Carlet determined the weight spectrum of $RM(m-5,m)$ $R M (m - 5, m)$ for $m\ge 10$ $m \geq 10$ using the Maiorana-McFarland construction, where he also studied the difficulties of trying to extend the results to $RM(m-6,m)$ $R M (m - 6, m)$ . In this paper, we fully determine the weight spectrum of $RM(m-6,m)$ $R M (m - 6, m)$ for $m\ge 12$ $m \geq 12$ , which gives a positive answer to an open question on the weight spectrum of $RM(m-c,m)$ $R M (m - c, m)$ for $c=6$ $c = 6$ . Moreover, we put forward a conjecture and verify it for some cases. If the conjecture is true, that open question can be completely solved.

Determining the weight spectrum of the Reed–Muller codes \(RM(m-6,m)\)

摘要