{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,13]],"date-time":"2025-09-13T15:39:20Z","timestamp":1757777960345,"version":"3.37.3"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T00:00:00Z","timestamp":1654214400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T00:00:00Z","timestamp":1654214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/N509449\/1"],"award-info":[{"award-number":["EP\/N509449\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2022,9]]},"abstract":"Abstract<\/jats:title>Let$${\\mathbb {Z}}_n = \\{Z_1, \\ldots , Z_n\\}$$<\/jats:tex-math>Z<\/mml:mi>n<\/mml:mi><\/mml:msub>=<\/mml:mo>{<\/mml:mo>Z<\/mml:mi>1<\/mml:mn><\/mml:msub>,<\/mml:mo>\u2026<\/mml:mo>,<\/mml:mo>Z<\/mml:mi>n<\/mml:mi><\/mml:msub>}<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>be a design; that is, a collection ofn<\/jats:italic>points$$Z_j \\in [-1,1]^d$$<\/jats:tex-math>Z<\/mml:mi>j<\/mml:mi><\/mml:msub>\u2208<\/mml:mo>[<\/mml:mo>-<\/mml:mo>1<\/mml:mn>,<\/mml:mo>1<\/mml:mn>]<\/mml:mo><\/mml:mrow>d<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We study the quality of quantisation of$$[-1,1]^d$$<\/jats:tex-math>[<\/mml:mo>-<\/mml:mo>1<\/mml:mn>,<\/mml:mo>1<\/mml:mn>]<\/mml:mo><\/mml:mrow>d<\/mml:mi><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>by the points of$${\\mathbb {Z}}_n$$<\/jats:tex-math>Z<\/mml:mi>n<\/mml:mi><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>and the problem of quality of coverage of$$[-1,1]^d$$<\/jats:tex-math>[<\/mml:mo>-<\/mml:mo>1<\/mml:mn>,<\/mml:mo>1<\/mml:mn>]<\/mml:mo><\/mml:mrow>d<\/mml:mi><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>by$${{{\\mathcal {B}}}}_d({\\mathbb {Z}}_n,r)$$<\/jats:tex-math>B<\/mml:mi>d<\/mml:mi><\/mml:msub>(<\/mml:mo>Z<\/mml:mi>n<\/mml:mi><\/mml:msub>,<\/mml:mo>r<\/mml:mi>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, the union of balls centred at$$Z_j \\in {\\mathbb {Z}}_n$$<\/jats:tex-math>Z<\/mml:mi>j<\/mml:mi><\/mml:msub>\u2208<\/mml:mo>Z<\/mml:mi>n<\/mml:mi><\/mml:msub><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We concentrate on the cases where the dimensiond<\/jats:italic>is not small,$$d\\ge 5$$<\/jats:tex-math>d<\/mml:mi>\u2265<\/mml:mo>5<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, andn<\/jats:italic>is not too large,$$n\\le 2^d$$<\/jats:tex-math>n<\/mml:mi>\u2264<\/mml:mo>2<\/mml:mn>d<\/mml:mi><\/mml:msup><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We define the design$${{\\mathbb {D}}_{n,\\delta }}$$<\/jats:tex-math>D<\/mml:mi>n<\/mml:mi>,<\/mml:mo>\u03b4<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>as a$$2^{d-1}$$<\/jats:tex-math>2<\/mml:mn>d<\/mml:mi>-<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>design defined on vertices of the cube$$[-\\delta ,\\delta ]^d$$<\/jats:tex-math>[<\/mml:mo>-<\/mml:mo>\u03b4<\/mml:mi>,<\/mml:mo>\u03b4<\/mml:mi>]<\/mml:mo><\/mml:mrow>d<\/mml:mi><\/mml:msup><\/mml:math><\/jats:alternatives><\/jats:inline-formula>,$$0\\le \\delta \\le 1$$<\/jats:tex-math>0<\/mml:mn>\u2264<\/mml:mo>\u03b4<\/mml:mi>\u2264<\/mml:mo>1<\/mml:mn><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. For this design, we derive a closed-form expression for the quantisation error and very accurate approximations for the coverage area$${\\text {vol}}{([-1,1]^d \\cap {{{\\mathcal {B}}}}_d({\\mathbb {Z}}_n,r))}$$<\/jats:tex-math>vol<\/mml:mtext>(<\/mml:mo>[<\/mml:mo>-<\/mml:mo>1<\/mml:mn>,<\/mml:mo>1<\/mml:mn>]<\/mml:mo><\/mml:mrow>d<\/mml:mi><\/mml:msup>\u2229<\/mml:mo>B<\/mml:mi>d<\/mml:mi><\/mml:msub>(<\/mml:mo>Z<\/mml:mi>n<\/mml:mi><\/mml:msub>,<\/mml:mo>r<\/mml:mi>)<\/mml:mo><\/mml:mrow>)<\/mml:mo><\/mml:mrow><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>. We provide results of a large-scale numerical investigation confirming the accuracy of the developed approximations and the efficiency of the designs\u00a0$${{\\mathbb {D}}_{n,\\delta }}$$<\/jats:tex-math>D<\/mml:mi>n<\/mml:mi>,<\/mml:mo>\u03b4<\/mml:mi><\/mml:mrow><\/mml:msub><\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00454-022-00396-7","type":"journal-article","created":{"date-parts":[[2022,6,3]],"date-time":"2022-06-03T14:03:20Z","timestamp":1654265000000},"page":"540-565","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Efficient Quantisation and Weak Covering of High Dimensional Cubes"],"prefix":"10.1007","volume":"68","author":[{"given":"Jack","family":"Noonan","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0630-8279","authenticated-orcid":false,"given":"Anatoly","family":"Zhigljavsky","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,6,3]]},"reference":[{"key":"396_CR1","doi-asserted-by":"crossref","unstructured":"Conway, J.H., Sloane, N.J.A.: Sphere Packings, Lattices and Groups. 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