MinT - Best Known (248−131, 248, s)-Nets in Base 3

Best Known (248−131, 248, s)-Nets in Base 3

(248−131, 248, 74)-Net over F₃ — Constructive and digital

Digital (117, 248, 74)-net over F₃, using

t-expansion [i] based on digital (107, 248, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(248−131, 248, 120)-Net over F₃ — Digital

Digital (117, 248, 120)-net over F₃, using

t-expansion [i] based on digital (113, 248, 120)-net over F₃, using
- net from sequence [i] based on digital (113, 119)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(248−131, 248, 751)-Net in Base 3 — Upper bound on s

There is no (117, 248, 752)-net in base 3, because

1 times m-reduction [i] would yield (117, 247, 752)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7197 116537 531918 754140 958252 116332 808678 765492 775195 967837 480569 827782 927626 660517 001915 006830 138850 592352 281875 951073 > 3²⁴⁷ [i]