Balance and Pattern Distribution of Sequences Derived from Pseudorandom Subsets of ℤ<sub> <i>q</i> </sub>

Abstract Let q be a positive integer and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mi>𝒮</m:mi> <m:mo>=</m:mo> <m:mo>{</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mo>⋯</m:mo> <m:mo>,</m:mo> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>T</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>}</m:mo> <m:mo>⊆</m:mo> <m:msub> <m:mi>ℤ</m:mi> <m:mi>q</m:mi> </m:msub> <m:mo>=</m:mo> <m:mo>{</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>}</m:mo> </m:mrow> </m:math> {\scr S} = \{{x_0},{x_1}, \cdots ,{x_{T - 1}}\}\subseteq {{\rm{\mathbb Z}}_q} = \{0,1, \ldots ,q - 1\} with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>0</m:mn> </m:msub> <m:mo>&lt;</m:mo> <m:msub> <m:mi>x</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>&lt;</m:mo> <m:mo>⋯</m:mo> <m:mo>&lt;</m:mo> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>T</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>1.</m:mn> </m:mrow> </m:math> 0 \le {x_0} &lt; {x_1} &lt;\cdots&lt; {x_{T - 1}} \le q - 1. . We derive from S three (finite) sequences: (1) For an integer M ≥ 2let ( s n )be the M -ary sequence defined by s n ≡ x n +1 − x n mod M, n =0 , 1,... ,T − 2 . (2) For an integer m ≥ 2let ( t n ) be the binary sequence defined by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:msub> <m:mi>s</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>≡</m:mo> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msub> <m:mtext> </m:mtext> <m:mi>mod</m:mi> <m:mtext> </m:mtext> <m:mi>M</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>⋯</m:mo> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo>−</m:mo> <m:mn>2.</m:mn> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \matrix{{{s_n} \equiv {x_{n + 1}} - {x_n}\,\bmod \,M,} &amp;#38; {n = 0,1, \cdots ,T - 2.}\cr} n =0 , 1,... ,T − 2 . (3) Let ( u n ) be the characteristic sequence of S , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:msub> <m:mi>t</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mtable columnalign="left"> <m:mtr columnalign="left"> <m:mtd columnalign="left"> <m:mn>1</m:mn> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mtext>if</m:mtext> <m:mtext> </m:mtext> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:msub> <m:mi>x</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mi>x</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>≤</m:mo> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr columnalign="left"> <m:mtd columnalign="left"> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mtext>otherwise</m:mtext> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>T</m:mi> <m:mo>−</m:mo> <m:mn>2.</m:mn> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \matrix{{{t_n} = \left\{{\matrix{1 \hfill &amp;#38; {{\rm{if}}\,1 \le {x_{n + 1}} - {x_n} \le m - 1,} \hfill\cr{0,} \hfill &amp;#38; {{\rm{otherwise}},} \hfill\cr}} \right.} &amp;#38; {n = 0,1, \ldots ,T - 2.}\cr} n =0 , 1,... ,q − 1. We study the balance and pattern distribution of the sequences ( s n ), ( t n )and ( u n ). For sets S with desirable pseudorandom properties, more precisely, sets with low correlation measures, we show the following: (1) The sequence ( s n ) is (asymptotically) balanced and has uniform pattern distribution if T is of smaller order of magnitude than q . (2) The sequence ( t n ) is balanced and has uniform pattern distribution if T is approximately <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mrow> <m:mtable> <m:mtr> <m:mtd> <m:mrow> <m:msub> <m:mi>u</m:mi> <m:mi>n</m:mi> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mtable columnalign="left"> <m:mtr columnalign="left"> <m:mtd columnalign="left"> <m:mn>1</m:mn> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mtext>if</m:mtext> <m:mtext> </m:mtext> <m:mi>n</m:mi> <m:mo>∈</m:mo> <m:mi>𝒮</m:mi> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> <m:mtr columnalign="left"> <m:mtd columnalign="left"> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mrow> </m:mtd> <m:mtd columnalign="left"> <m:mrow> <m:mtext>otherwise</m:mtext> <m:mo>,</m:mo> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mrow> </m:mrow> </m:mtd> <m:mtd> <m:mrow> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>…</m:mo> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo>−</m:mo> <m:mn>1.</m:mn> </m:mrow> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:math> \matrix{{{u_n} = \left\{{\matrix{1 \hfill &amp;#38; {{\rm{if}}\,n \in {\scr S},} \hfill\cr{0,} \hfill &amp;#38; {{\rm{otherwise}},} \hfill\cr}} \right.} &amp;#38; {n = 0,1, \ldots ,q - 1.}\cr} . (3) The sequence ( u n ) is balanced and has uniform pattern distribution if T is approximately q 2 . 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.