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<p>Does every three-word code have a finite completion? Up to now, this famous question in the theory of codes remains open. Motivated by this problem, we construct several types of three-word codes with the form <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="236_2025_487_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="92" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{a,\ aba,\ u\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mspace width="4pt" /> <mi>a</mi> <mi>b</mi> <mi>a</mi> <mo>,</mo> <mspace width="4pt" /> <mi>u</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="236_2025_487_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="83" /> </InlineMediaObject> <EquationSource Format="TEX">\(\{a,\ ab,\ v\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">{</mo> <mi>a</mi> <mo>,</mo> <mspace width="4pt" /> <mi>a</mi> <mi>b</mi> <mo>,</mo> <mspace width="4pt" /> <mi>v</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> which have finite completions.</p>

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Three-word codes \(\{a,\ aba,\ u\}\) and \(\{a,\ ab,\ v\}\) having finite completions

  • 期刊论文

  • 发表日期：

    2025年4月16日

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  • Chunhua Cao,
  • Lei Liao,
  • Zhongmei Yan,
  • Di Yang,
  • Yuguang Yuan

摘要

Does every three-word code have a finite completion? Up to now, this famous question in the theory of codes remains open. Motivated by this problem, we construct several types of three-word codes with the form \(\{a,\ aba,\ u\}\) { a , a b a , u } and \(\{a,\ ab,\ v\}\) { a , a b , v } which have finite completions.