{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:43:31Z","timestamp":1776797011517,"version":"3.51.2"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"287","license":[{"start":{"date-parts":[[2014,7,24]],"date-time":"2014-07-24T00:00:00Z","timestamp":1406160000000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    For lengths\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"8\">\n                        <mml:semantics>\n                          <mml:mn>8<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">8<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"16\">\n                        <mml:semantics>\n                          <mml:mn>16<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">16<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"24\">\n                        <mml:semantics>\n                          <mml:mn>24<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">24<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , it is known that there is an extremal Type\u00a0II\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z Subscript 2 k\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}_{2k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -code for every positive integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . In this paper, we show that there is an extremal Type\u00a0II\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z Subscript 2 k\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>2<\/mml:mn>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}_{2k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -code of lengths\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"32 comma 40 comma 48 comma 56\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mn>32<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>40<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>48<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mn>56<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">32,40,48,56<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"64\">\n                        <mml:semantics>\n                          <mml:mn>64<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">64<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for every positive integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . For length\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"72\">\n                        <mml:semantics>\n                          <mml:mn>72<\/mml:mn>\n                          <mml:annotation encoding=\"application\/x-tex\">72<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , it is also shown that there is an extremal Type\u00a0II\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Z Subscript 4 k\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Z<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mn>4<\/mml:mn>\n                              <mml:mi>k<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Z}_{4k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -code for every positive integer\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    with\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k greater-than-or-equal-to 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k \\ge 2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/s0025-5718-2013-02750-0","type":"journal-article","created":{"date-parts":[[2013,7,24]],"date-time":"2013-07-24T15:04:14Z","timestamp":1374678254000},"page":"1427-1446","source":"Crossref","is-referenced-by-count":3,"title":["On the existence of extremal Type II \u2124_{2\ud835\udd5c}-codes"],"prefix":"10.1090","volume":"83","author":[{"given":"Masaaki","family":"Harada","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tsuyoshi","family":"Miezaki","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,7,24]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"1392","DOI":"10.1016\/j.ejc.2008.11.007","article-title":"A survey on spherical designs and algebraic combinatorics on spheres","volume":"30","author":"Bannai, Eiichi","year":"2009","journal-title":"European J. 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