<p>A pair of sequences is called a Golay complementary pair (GCP), when it has zero aperiodic auto-correlation function (AACF) sum at every non-zero time shift. Due to its low peak-to-mean envelope power ratio (PMEPR), which is bounded by 2, it is used in many wireless communication applications. In this paper, we present direct constructions of 4<i>h</i>-ary <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_782_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">$(h\ge 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> GCPs of lengths of the forms <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_782_Article_IEq2.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$3\times 2^{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>×</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_782_Article_IEq3.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$11\times 2^{m}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>11</mn> <mo>×</mo> <msup> <mn>2</mn> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12095_2025_782_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$(m\ge 1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mo>≥</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> using multi-variable extended Boolean functions (EBFs). We have also provided the generating functions of the complementary mates for the proposed GCPs.</p>

Constructions of non-binary Golay complementary pairs of new lengths

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发表日期：

2025年2月14日

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Piyush Priyanshu,
Abhishek Roy,
Subhabrata Paul,
Sudhan Majhi

摘要

A pair of sequences is called a Golay complementary pair (GCP), when it has zero aperiodic auto-correlation function (AACF) sum at every non-zero time shift. Due to its low peak-to-mean envelope power ratio (PMEPR), which is bounded by 2, it is used in many wireless communication applications. In this paper, we present direct constructions of 4h-ary $(h\ge 1)$ $(h \geq 1)$ GCPs of lengths of the forms $3\times 2^{m}$ $3 \times 2^{m}$ and $11\times 2^{m}$ $11 \times 2^{m}$ , $(m\ge 1)$ $(m \geq 1)$ using multi-variable extended Boolean functions (EBFs). We have also provided the generating functions of the complementary mates for the proposed GCPs.