<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>0120-419X</journal-id>
<journal-title><![CDATA[Revista Integración]]></journal-title>
<abbrev-journal-title><![CDATA[Integración - UIS]]></abbrev-journal-title>
<issn>0120-419X</issn>
<publisher>
<publisher-name><![CDATA[Universidad Industrial de Santander]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S0120-419X2021000100005</article-id>
<article-id pub-id-type="doi">10.18273/revint.v39n1-2021005</article-id>
<title-group>
<article-title xml:lang="es"><![CDATA[Imágenes de Gray de códigos consta-cíclicos sobre anillos de Galois R de índice de nilpotencia 3]]></article-title>
<article-title xml:lang="en"><![CDATA[Gray images of constacyclic codes over Galois rings R of nilpotency index 3]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[García-Ramírez]]></surname>
<given-names><![CDATA[Angel R.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[López-Andrade]]></surname>
<given-names><![CDATA[Carlos A.]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Villa-Hernández]]></surname>
<given-names><![CDATA[David]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Benemérita Universidad Autónoma de Puebla Facultad de Ciencias Físico Matemáticas ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>México</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Benemérita Universidad Autónoma de Puebla Facultad de Ciencias Físico Matemáticas ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>México</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Benemérita Universidad Autónoma de Puebla Facultad de Ciencias Físico Matemáticas ]]></institution>
<addr-line><![CDATA[ ]]></addr-line>
<country>México</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>06</month>
<year>2021</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>06</month>
<year>2021</year>
</pub-date>
<volume>39</volume>
<numero>1</numero>
<fpage>57</fpage>
<lpage>78</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_arttext&amp;pid=S0120-419X2021000100005&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_abstract&amp;pid=S0120-419X2021000100005&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.org.co/scielo.php?script=sci_pdf&amp;pid=S0120-419X2021000100005&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen Estableceremos condiciones necesarias y suficientes para que la imagen bajo la función de Gray de un R-código consta-cíclico sea un pm-código cuasi-cíclico. Estudiamos el anillo de vectores de Witt para obtener una manera de operar las µ-reducciones de las componentes p-ádicas de los elementos de los anillos de Galois de índice de nilpotencia 3, R = GR (p3 , m). Analizamos a los anillos de Galois, sus propiedades más relevantes, y en particular la representación p-ádica de sus elementos. Más adelante, examinamos la construcción del anillo de vectores de Witt y sus operaciones, en particular, obtenemos expresiones explícitas para las operaciones de suma y producto de los elementos en el anillo truncado de vectores de Witt de longitud 3, W3 (pm). Finalmente, utilizamos las operaciones de éstos últimos y un isomorfismo entre GR (p3 , m) y W3 (pm) para operar las µ-reducciones antes descritas.]]></p></abstract>
<abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract We will state necessary and sufficient conditions for the image under the Gray map of a R-constacyclic code to bepm-quasi-cyclic code. We study the Witt vectors to get a way to operate the µ-reduction of padic components of the elements of the Galois rings of nilpotency index 3, R = GR (p3 , m). We analyze Galois rings, its mostly relevant properties, and we focus in the p-adic representation of their elements. Later on, we examine construction of the Witt vectors rings and its operations, in particular, we get explicit expressions for operations of addition and product of the elements in the truncated Witt vectors ring of length 3, W3 (pm). Finally, we will use these operations and an isomorphism between GR (p3 , m) and W3 (pm) to get a way to operate the µ-reductions described above.]]></p></abstract>
<kwd-group>
<kwd lng="es"><![CDATA[Anillo de vectores de Witt]]></kwd>
<kwd lng="es"><![CDATA[anillos de Galois]]></kwd>
<kwd lng="es"><![CDATA[representación p-ádica]]></kwd>
<kwd lng="es"><![CDATA[códigos consta-cíclicos]]></kwd>
<kwd lng="es"><![CDATA[códigos cuasi-cíclicos]]></kwd>
<kwd lng="es"><![CDATA[MSC2010: 13F35]]></kwd>
<kwd lng="es"><![CDATA[16P10]]></kwd>
<kwd lng="es"><![CDATA[12F10]]></kwd>
<kwd lng="es"><![CDATA[94B60]]></kwd>
<kwd lng="en"><![CDATA[Witt&#8217;s vectors ring]]></kwd>
<kwd lng="en"><![CDATA[Galois rings]]></kwd>
<kwd lng="en"><![CDATA[p-adic representation]]></kwd>
<kwd lng="en"><![CDATA[constacyclic codes]]></kwd>
<kwd lng="en"><![CDATA[quasi-cyclic codes]]></kwd>
</kwd-group>
</article-meta>
</front><back>
<ref-list>
<ref id="B1">
<label>[1]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Dinh]]></surname>
<given-names><![CDATA[H.Q.]]></given-names>
</name>
<name>
<surname><![CDATA[Liu]]></surname>
<given-names><![CDATA[H.]]></given-names>
</name>
<name>
<surname><![CDATA[Liu]]></surname>
<given-names><![CDATA[X.]]></given-names>
</name>
<name>
<surname><![CDATA[Sriboonchitta]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On structure and distances of some classes of repeated-root constacyclic codes over Galois rings]]></article-title>
<source><![CDATA[Finite Fields Appl]]></source>
<year>2017</year>
<volume>43</volume>
<page-range>86-105</page-range></nlm-citation>
</ref>
<ref id="B2">
<label>[2]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Dinh]]></surname>
<given-names><![CDATA[H.Q.]]></given-names>
</name>
<name>
<surname><![CDATA[López-Permouth]]></surname>
<given-names><![CDATA[S.R.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Cyclic and negacyclic codes over finite chain rings]]></article-title>
<source><![CDATA[IEEE Trans. Inform. Theory]]></source>
<year>2004</year>
<volume>50</volume>
<numero>8</numero>
<issue>8</issue>
<page-range>1728-44</page-range></nlm-citation>
</ref>
<ref id="B3">
<label>[3]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Gómez-Calderon]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
<name>
<surname><![CDATA[Mullen]]></surname>
<given-names><![CDATA[G.L.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Galois rings and algebraic cryptography]]></article-title>
<source><![CDATA[Acta Arith]]></source>
<year>1991</year>
<volume>59</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>317-28</page-range></nlm-citation>
</ref>
<ref id="B4">
<label>[4]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Greferath]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
<name>
<surname><![CDATA[Schmidt]]></surname>
<given-names><![CDATA[S.E.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Gray isometries for finite chain rings and a nonlinear ternary (36, 3 12 , 15) code]]></article-title>
<source><![CDATA[IEEE Trans. Inform. Theory]]></source>
<year>1999</year>
<volume>45</volume>
<numero>7</numero>
<issue>7</issue>
<page-range>2522-4</page-range></nlm-citation>
</ref>
<ref id="B5">
<label>[5]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Nechaev]]></surname>
<given-names><![CDATA[A.A.]]></given-names>
</name>
<name>
<surname><![CDATA[Khonol&#8217;d]]></surname>
<given-names><![CDATA[T.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Fully weighted modules and representations of codes]]></article-title>
<source><![CDATA[Probl. Inf. Transm]]></source>
<year>1999</year>
<volume>35</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>205-23</page-range></nlm-citation>
</ref>
<ref id="B6">
<label>[6]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Jacobson]]></surname>
<given-names><![CDATA[N.]]></given-names>
</name>
</person-group>
<source><![CDATA[Basic Algebra II]]></source>
<year>2009</year>
<edition>2nd ed</edition>
<publisher-loc><![CDATA[New York ]]></publisher-loc>
<publisher-name><![CDATA[Dover Publications]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B7">
<label>[7]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Hammons]]></surname>
<given-names><![CDATA[Jr. A.R.]]></given-names>
</name>
<name>
<surname><![CDATA[Kumar]]></surname>
<given-names><![CDATA[P.V.]]></given-names>
</name>
<name>
<surname><![CDATA[Calderbank]]></surname>
<given-names><![CDATA[A.R.]]></given-names>
</name>
<name>
<surname><![CDATA[Sloane]]></surname>
<given-names><![CDATA[N.J.A.]]></given-names>
</name>
<name>
<surname><![CDATA[Solé]]></surname>
<given-names><![CDATA[P.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[The Z4-linearity of Kerdock, Preparata, Goethals, and related codes]]></article-title>
<source><![CDATA[IEEE Trans. Inform. Tehory]]></source>
<year>1994</year>
<volume>40</volume>
<numero>2</numero>
<issue>2</issue>
<page-range>301-19</page-range></nlm-citation>
</ref>
<ref id="B8">
<label>[8]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Kanwar]]></surname>
<given-names><![CDATA[P.]]></given-names>
</name>
<name>
<surname><![CDATA[López-Permouth]]></surname>
<given-names><![CDATA[S.R.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Cyclic Codes over the Integers Modulo pm]]></article-title>
<source><![CDATA[Finite Fields Appl]]></source>
<year>1997</year>
<volume>3</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>334-52</page-range></nlm-citation>
</ref>
<ref id="B9">
<label>[9]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Karpilovsky]]></surname>
<given-names><![CDATA[G.]]></given-names>
</name>
</person-group>
<source><![CDATA[Topics in field theory]]></source>
<year>1989</year>
<volume>155</volume>
<publisher-loc><![CDATA[Amsterdam ]]></publisher-loc>
<publisher-name><![CDATA[North-Holland Publishing Co]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B10">
<label>[10]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Lidl]]></surname>
<given-names><![CDATA[R.]]></given-names>
</name>
<name>
<surname><![CDATA[Niederreiter]]></surname>
<given-names><![CDATA[H.]]></given-names>
</name>
</person-group>
<source><![CDATA[Finite fields]]></source>
<year>1997</year>
<volume>20</volume>
<edition>2nd ed</edition>
<publisher-loc><![CDATA[Cambridge ]]></publisher-loc>
<publisher-name><![CDATA[Cambridge University Press]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B11">
<label>[11]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Ling]]></surname>
<given-names><![CDATA[S.]]></given-names>
</name>
<name>
<surname><![CDATA[Blackford]]></surname>
<given-names><![CDATA[J.T.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Zpk+1-Linear Codes]]></article-title>
<source><![CDATA[IEEE Trans. Inform. Theory]]></source>
<year>2002</year>
<volume>48</volume>
<numero>9</numero>
<issue>9</issue>
<page-range>2592-605</page-range></nlm-citation>
</ref>
<ref id="B12">
<label>[12]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[López-Andrade]]></surname>
<given-names><![CDATA[C.A.]]></given-names>
</name>
<name>
<surname><![CDATA[Tapia-Recillas]]></surname>
<given-names><![CDATA[H.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[On the linearity and quasi-cyclicity of the gray image of codes over a Galois ring]]></article-title>
<source><![CDATA[Amer. Math. Soc]]></source>
<year>2011</year>
<volume>537</volume>
<page-range>255-68</page-range></nlm-citation>
</ref>
<ref id="B13">
<label>[13]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Nechaev]]></surname>
<given-names><![CDATA[A.A.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Kerdock code in a cyclic form]]></article-title>
<source><![CDATA[Discrete Math. Appl]]></source>
<year>1991</year>
<volume>1</volume>
<numero>4</numero>
<issue>4</issue>
<page-range>365-84</page-range></nlm-citation>
</ref>
<ref id="B14">
<label>[14]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Özbudak]]></surname>
<given-names><![CDATA[F.]]></given-names>
</name>
<name>
<surname><![CDATA[Saygi]]></surname>
<given-names><![CDATA[Z.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Some constructions of systematic authentication codes using galois rings]]></article-title>
<source><![CDATA[Des. Codes Cryptogr]]></source>
<year>2006</year>
<volume>41</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>343-57</page-range></nlm-citation>
</ref>
<ref id="B15">
<label>[15]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Sobhani]]></surname>
<given-names><![CDATA[R.]]></given-names>
</name>
<name>
<surname><![CDATA[Esmaeili]]></surname>
<given-names><![CDATA[M.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Cyclic and negacyclic codes over the Galois ring GR(p2 , m)]]></article-title>
<source><![CDATA[Discrete Appl. Math]]></source>
<year>2009</year>
<volume>157</volume>
<numero>13</numero>
<issue>13</issue>
<page-range>2892-903</page-range></nlm-citation>
</ref>
<ref id="B16">
<label>[16]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Shanbhag]]></surname>
<given-names><![CDATA[A.G.]]></given-names>
</name>
<name>
<surname><![CDATA[Kumar]]></surname>
<given-names><![CDATA[P.V.]]></given-names>
</name>
<name>
<surname><![CDATA[Helleseth]]></surname>
<given-names><![CDATA[T.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[An upper bound for the extended Kloosterman sums over Galois rings]]></article-title>
<source><![CDATA[Finite Fields Appl]]></source>
<year>1998</year>
<volume>4</volume>
<numero>3</numero>
<issue>3</issue>
<page-range>218-38</page-range></nlm-citation>
</ref>
<ref id="B17">
<label>[17]</label><nlm-citation citation-type="book">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Wan]]></surname>
<given-names><![CDATA[Z.X.]]></given-names>
</name>
</person-group>
<source><![CDATA[Lectures on finite fields and Galois rings]]></source>
<year>2003</year>
<publisher-loc><![CDATA[River Edge, NJ ]]></publisher-loc>
<publisher-name><![CDATA[World Scientific Publishing Co., Inc]]></publisher-name>
</nlm-citation>
</ref>
<ref id="B18">
<label>[18]</label><nlm-citation citation-type="journal">
<person-group person-group-type="author">
<name>
<surname><![CDATA[Wolfmann]]></surname>
<given-names><![CDATA[J.]]></given-names>
</name>
</person-group>
<article-title xml:lang=""><![CDATA[Binary images of cyclic codes over Z4]]></article-title>
<source><![CDATA[IEEE Trans. Inform. Theory]]></source>
<year>2001</year>
<volume>47</volume>
<numero>5</numero>
<issue>5</issue>
<page-range>1773-9</page-range></nlm-citation>
</ref>
</ref-list>
</back>
</article>
