MinT - Best Known (109, 109+135, s)-Nets in Base 3

Best Known (109, 109+135, s)-Nets in Base 3

(109, 109+135, 74)-Net over F₃ — Constructive and digital

Digital (109, 244, 74)-net over F₃, using

t-expansion [i] based on digital (107, 244, 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]

(109, 109+135, 104)-Net over F₃ — Digital

Digital (109, 244, 104)-net over F₃, using

t-expansion [i] based on digital (102, 244, 104)-net over F₃, using
- net from sequence [i] based on digital (102, 103)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 102 and N(F) ≥ 104, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(109, 109+135, 629)-Net in Base 3 — Upper bound on s

There is no (109, 244, 630)-net in base 3, because

1 times m-reduction [i] would yield (109, 243, 630)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 94 884909 901655 328972 920024 636497 285441 345305 981569 849736 672199 511659 887751 260984 367992 023938 857557 151754 913107 777953 > 3²⁴³ [i]