{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,10,10]],"date-time":"2022-10-10T14:41:17Z","timestamp":1665412877676},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2020,10,11]],"date-time":"2020-10-11T00:00:00Z","timestamp":1602374400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,10,11]],"date-time":"2020-10-11T00:00:00Z","timestamp":1602374400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2022,11]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Traditional global stability measure for sequences is hard to determine because of large search space. We propose the <jats:italic>k<\/jats:italic>-error linear complexity with a zone restriction for measuring the local stability of sequences. For several classes of sequences, we demonstrate that the <jats:italic>k<\/jats:italic>-error linear complexity is identical to the <jats:italic>k<\/jats:italic>-error linear complexity within a zone, while the length of a zone is much smaller than the whole period when the <jats:italic>k<\/jats:italic>-error linear complexity is large. These sequences have periods <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, or <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^v r$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>v<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:mi>r<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (<jats:italic>r<\/jats:italic> odd prime and 2 is primitive modulo <jats:italic>r<\/jats:italic>), or <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^v p_1^{s_1} \\cdots p_n^{s_n}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msup>\n                      <mml:mn>2<\/mml:mn>\n                      <mml:mi>v<\/mml:mi>\n                    <\/mml:msup>\n                    <mml:msubsup>\n                      <mml:mi>p<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                      <mml:msub>\n                        <mml:mi>s<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                    <\/mml:msubsup>\n                    <mml:mo>\u22ef<\/mml:mo>\n                    <mml:msubsup>\n                      <mml:mi>p<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:msub>\n                        <mml:mi>s<\/mml:mi>\n                        <mml:mi>n<\/mml:mi>\n                      <\/mml:msub>\n                    <\/mml:msubsup>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> (<jats:inline-formula><jats:alternatives><jats:tex-math>$$p_i$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mi>i<\/mml:mi>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is an odd prime and 2 is primitive modulo <jats:inline-formula><jats:alternatives><jats:tex-math>$$p_i^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msubsup>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mi>i<\/mml:mi>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msubsup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$1\\le i \\le n$$<\/jats:tex-math><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>i<\/mml:mi>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>) respectively. In particular, we completely determine the spectrum of 1-error linear complexity with any zone length for an arbitrary <jats:inline-formula><jats:alternatives><jats:tex-math>$$2^n$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mn>2<\/mml:mn>\n                    <mml:mi>n<\/mml:mi>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-periodic binary sequence.<\/jats:p>","DOI":"10.1007\/s00200-020-00467-3","type":"journal-article","created":{"date-parts":[[2020,10,11]],"date-time":"2020-10-11T15:03:15Z","timestamp":1602428595000},"page":"485-504","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the stability of periodic binary sequences with zone restriction"],"prefix":"10.1007","volume":"33","author":[{"given":"Ming","family":"Su","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qiang","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,10,11]]},"reference":[{"key":"467_CR1","doi-asserted-by":"crossref","unstructured":"Alecu, A., S\u0103l\u0103gean, A.: An approximation algorithm for computing the $$k$$-error linear complexity of sequences using the discrete fourier transform. 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