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Codes Cryptogr."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    An (\n                    <jats:italic>n<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>R<\/jats:italic>\n                    )-covering sequence is a cyclic sequence whose consecutive\n                    <jats:italic>n<\/jats:italic>\n                    -tuples form a code of length\n                    <jats:italic>n<\/jats:italic>\n                    and covering radius\n                    <jats:italic>R<\/jats:italic>\n                    . Using several construction methods improvements of the upper bounds on the length of such sequences for\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$n \\le 20$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mn>20<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$1 \\le R \\le 3$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mi>R<\/mml:mi>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    , are obtained. The definition is generalized in two directions. An (\n                    <jats:italic>n<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>m<\/jats:italic>\n                    ,\u00a0\n                    <jats:italic>R<\/jats:italic>\n                    )-covering sequence code is a set of cyclic sequences of length\n                    <jats:italic>m<\/jats:italic>\n                    whose consecutive\n                    <jats:italic>n<\/jats:italic>\n                    -tuples form a code of length\u00a0\n                    <jats:italic>n<\/jats:italic>\n                    and covering radius\n                    <jats:italic>R<\/jats:italic>\n                    . The definition is also generalized to arrays in which the\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$m \\times n$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>m<\/mml:mi>\n                            <mml:mo>\u00d7<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    sub-matrices form a covering code with covering radius\n                    <jats:italic>R<\/jats:italic>\n                    . We prove that asymptotically there are covering sequences that attain the sphere-covering bound up to a constant factor.\n                  <\/jats:p>","DOI":"10.1007\/s10623-025-01726-5","type":"journal-article","created":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T06:55:12Z","timestamp":1758783312000},"page":"5445-5471","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Constructions of covering sequences and 2D-sequences"],"prefix":"10.1007","volume":"93","author":[{"given":"Yeow Meng","family":"Chee","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tuvi","family":"Etzion","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hoang","family":"Ta","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Van Khu","family":"Vu","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,25]]},"reference":[{"key":"1726_CR1","doi-asserted-by":"crossref","unstructured":"Berkowitz R., Kopparty S.: Robust positioning patterns. 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