{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,5]],"date-time":"2025-12-05T21:21:20Z","timestamp":1764969680218,"version":"3.46.0"},"reference-count":37,"publisher":"Association for Computing Machinery (ACM)","issue":"6","funder":[{"name":"Guangdong S&T Program","award":["2023B1515120026"],"award-info":[{"award-number":["2023B1515120026"]}]},{"DOI":"10.13039\/501100001809","name":"NSFC","doi-asserted-by":"crossref","award":["U21B2023"],"award-info":[{"award-number":["U21B2023"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]},{"name":"DEGP Innovation Team","award":["2022KCXTD025"],"award-info":[{"award-number":["2022KCXTD025"]}]},{"name":"Shenzhen Science and Technology Program","award":["KJZD20240903100022028","KQTD20210811090044003","RCJC20200714114435012"],"award-info":[{"award-number":["KJZD20240903100022028","KQTD20210811090044003","RCJC20200714114435012"]}]}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Trans. Graph."],"published-print":{"date-parts":[[2025,12]]},"abstract":"<jats:p>\n                    Low-discrepancy sequences have seen widespread adoption in computer graphics thanks to the superior rates of convergence that they provide. Because rendering integrals often are comprised of products of lower-dimensional integrals, recent work has focused on developing sequences that are also well-distributed in lower-dimensional projections. To this end, we introduce a novel construction of binary-based (0, 4)-sequences; that is, progressive fully multi-stratified sequences of 4D points, and extend the idea to higher power-of-two dimensions. We further show that not only it is possible to nest lower-dimensional sequences in higher-dimensional ones\u2014for example, embedding a (0, 2)-sequence within our (0, 4)-sequence\u2014but that we can ensemble two (0, 2)-sequences into a (0, 4)-sequence, four (0, 4)-sequences into a (0,16)-sequence, and so on. Such sequences can provide excellent rates of convergence when integrals include lower-dimensional integration problems in 2, 4, 16,... dimensions. Our construction is based on using 2\u00d72 block matrices as symbols to construct larger matrices that potentially generate a sequence with the target (0,\n                    <jats:italic toggle=\"yes\">s<\/jats:italic>\n                    )-sequence in base\n                    <jats:italic toggle=\"yes\">s<\/jats:italic>\n                    property. We describe how to search for suitable alphabets and identify two distinct, cross-related alphabets of block symbols, which we call\n                    <jats:italic toggle=\"yes\">s<\/jats:italic>\n                    and\n                    <jats:italic toggle=\"yes\">z<\/jats:italic>\n                    , hence\n                    <jats:italic toggle=\"yes\">SZ<\/jats:italic>\n                    for the resulting family of sequences. Given the alphabets, we construct candidate generator matrices and search for valid sets of matrices. We then infer a simple recurrence formula to construct full-resolution (64-bit) matrices. Because our generator matrices are binary, they allow highly-efficient implementation using bitwise operations and can be used as a drop-in replacement for Sobol matrices in existing applications. We compare SZ sequences to state-of-the-art low discrepancy sequences, and demonstrate mean relative squared error improvements up to 1.93\u00d7 in common rendering applications.\n                  <\/jats:p>","DOI":"10.1145\/3763272","type":"journal-article","created":{"date-parts":[[2025,12,4]],"date-time":"2025-12-04T17:15:39Z","timestamp":1764868539000},"page":"1-14","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["SZ Sequences: Binary-Based (0, 2\n                    <i>\n                      <sup>q<\/sup>\n                    <\/i>\n                    )-Sequences"],"prefix":"10.1145","volume":"44","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-2348-6897","authenticated-orcid":false,"given":"Abdalla G. M.","family":"Ahmed","sequence":"first","affiliation":[{"name":"Shenzhen University, Shenzhen, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0566-8291","authenticated-orcid":false,"given":"Matt","family":"Pharr","sequence":"additional","affiliation":[{"name":"NVIDIA, San Francisco, USA"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-3123-9388","authenticated-orcid":false,"given":"Victor","family":"Ostromoukhov","sequence":"additional","affiliation":[{"name":"Universit\u00e9 Claude Bernard Lyon, Lyon, France"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3212-0544","authenticated-orcid":false,"given":"Hui","family":"Huang","sequence":"additional","affiliation":[{"name":"Shenzhen University, Shenzhen, China"}]}],"member":"320","published-online":{"date-parts":[[2025,12,4]]},"reference":[{"key":"e_1_2_1_1_1","doi-asserted-by":"publisher","DOI":"10.2312\/sr.20241147"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.2312\/cgvc.20251215"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1145\/2980179.2980218"},{"key":"e_1_2_1_4_1","doi-asserted-by":"publisher","DOI":"10.1145\/3618308"},{"key":"e_1_2_1_5_1","doi-asserted-by":"publisher","DOI":"10.1145\/3414685.3417881"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1145\/3450626.3459880"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1145\/3406183"},{"key":"e_1_2_1_8_1","doi-asserted-by":"publisher","unstructured":"Pontus Andersson Jim Nilsson Peter Shirley and Tomas Akenine-M\u00f6ller. 2021. 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