{"title":"伪随机子集衍生序列的平衡与模式分布","authors":"Huaning Liu, Arne Winterhof","doi":"10.2478/udt-2021-0009","DOIUrl":null,"url":null,"abstract":"Abstract Let q be a positive integer and 𝒮={x0,x1,⋯,xT−1}⊆ℤq={0,1,…,q−1} {\\scr S} = \\{{x_0},{x_1}, \\cdots ,{x_{T - 1}}\\}\\subseteq {{\\rm{\\mathbb Z}}_q} = \\{0,1, \\ldots ,q - 1\\} with 0≤x0<x1<⋯<xT−1≤q−1. 0 \\le {x_0} < {x_1} <\\cdots< {x_{T - 1}} \\le q - 1. . We derive from S three (finite) sequences: (1) For an integer M ≥ 2let (sn)be the M-ary sequence defined by sn ≡ xn+1 − xn mod M, n =0, 1,...,T − 2. (2) For an integer m ≥ 2let (tn) be the binary sequence defined by sn≡xn+1−xn mod M,n=0,1,⋯,T−2. \\matrix{{{s_n} \\equiv {x_{n + 1}} - {x_n}\\,\\bmod \\,M,} & {n = 0,1, \\cdots ,T - 2.}\\cr} n =0, 1,...,T − 2. (3) Let (un) be the characteristic sequence of S, tn={1if 1≤xn+1−xn≤m−1,0,otherwise,n=0,1,…,T−2. \\matrix{{{t_n} = \\left\\{{\\matrix{1 \\hfill & {{\\rm{if}}\\,1 \\le {x_{n + 1}} - {x_n} \\le m - 1,} \\hfill\\cr{0,} \\hfill & {{\\rm{otherwise}},} \\hfill\\cr}} \\right.} & {n = 0,1, \\ldots ,T - 2.}\\cr} n =0, 1,...,q − 1. We study the balance and pattern distribution of the sequences (sn), (tn)and (un). For sets S with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following: (1) The sequence (sn) is (asymptotically) balanced and has uniform pattern distribution if T is of smaller order of magnitude than q. (2) The sequence (tn) is balanced and has uniform pattern distribution if T is approximately un={1if n∈𝒮,0,otherwise,n=0,1,…,q−1. \\matrix{{{u_n} = \\left\\{{\\matrix{1 \\hfill & {{\\rm{if}}\\,n \\in {\\scr S},} \\hfill\\cr{0,} \\hfill & {{\\rm{otherwise}},} \\hfill\\cr}} \\right.} & {n = 0,1, \\ldots ,q - 1.}\\cr} . (3) The sequence (un) is balanced and has uniform pattern distribution if T is approximately q2. These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.","PeriodicalId":23390,"journal":{"name":"Uniform distribution theory","volume":"12 1","pages":"89 - 108"},"PeriodicalIF":0.0000,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤq\",\"authors\":\"Huaning Liu, Arne Winterhof\",\"doi\":\"10.2478/udt-2021-0009\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Let q be a positive integer and 𝒮={x0,x1,⋯,xT−1}⊆ℤq={0,1,…,q−1} {\\\\scr S} = \\\\{{x_0},{x_1}, \\\\cdots ,{x_{T - 1}}\\\\}\\\\subseteq {{\\\\rm{\\\\mathbb Z}}_q} = \\\\{0,1, \\\\ldots ,q - 1\\\\} with 0≤x0<x1<⋯<xT−1≤q−1. 0 \\\\le {x_0} < {x_1} <\\\\cdots< {x_{T - 1}} \\\\le q - 1. . We derive from S three (finite) sequences: (1) For an integer M ≥ 2let (sn)be the M-ary sequence defined by sn ≡ xn+1 − xn mod M, n =0, 1,...,T − 2. (2) For an integer m ≥ 2let (tn) be the binary sequence defined by sn≡xn+1−xn mod M,n=0,1,⋯,T−2. \\\\matrix{{{s_n} \\\\equiv {x_{n + 1}} - {x_n}\\\\,\\\\bmod \\\\,M,} & {n = 0,1, \\\\cdots ,T - 2.}\\\\cr} n =0, 1,...,T − 2. (3) Let (un) be the characteristic sequence of S, tn={1if 1≤xn+1−xn≤m−1,0,otherwise,n=0,1,…,T−2. \\\\matrix{{{t_n} = \\\\left\\\\{{\\\\matrix{1 \\\\hfill & {{\\\\rm{if}}\\\\,1 \\\\le {x_{n + 1}} - {x_n} \\\\le m - 1,} \\\\hfill\\\\cr{0,} \\\\hfill & {{\\\\rm{otherwise}},} \\\\hfill\\\\cr}} \\\\right.} & {n = 0,1, \\\\ldots ,T - 2.}\\\\cr} n =0, 1,...,q − 1. We study the balance and pattern distribution of the sequences (sn), (tn)and (un). For sets S with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following: (1) The sequence (sn) is (asymptotically) balanced and has uniform pattern distribution if T is of smaller order of magnitude than q. (2) The sequence (tn) is balanced and has uniform pattern distribution if T is approximately un={1if n∈𝒮,0,otherwise,n=0,1,…,q−1. \\\\matrix{{{u_n} = \\\\left\\\\{{\\\\matrix{1 \\\\hfill & {{\\\\rm{if}}\\\\,n \\\\in {\\\\scr S},} \\\\hfill\\\\cr{0,} \\\\hfill & {{\\\\rm{otherwise}},} \\\\hfill\\\\cr}} \\\\right.} & {n = 0,1, \\\\ldots ,q - 1.}\\\\cr} . (3) The sequence (un) is balanced and has uniform pattern distribution if T is approximately q2. These results are motivated by earlier results for the sets of quadratic residues and primitive roots modulo a prime. We unify these results and derive many further (asymptotically) balanced sequences with uniform pattern distribution from pseudorandom subsets.\",\"PeriodicalId\":23390,\"journal\":{\"name\":\"Uniform distribution theory\",\"volume\":\"12 1\",\"pages\":\"89 - 108\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2021-12-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Uniform distribution theory\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2478/udt-2021-0009\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Uniform distribution theory","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2478/udt-2021-0009","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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