MinT - Best Known (115, 115+123, s)-Nets in Base 3

Best Known (115, 115+123, s)-Nets in Base 3

(115, 115+123, 74)-Net over F₃ — Constructive and digital

Digital (115, 238, 74)-net over F₃, using

t-expansion [i] based on digital (107, 238, 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]

(115, 115+123, 120)-Net over F₃ — Digital

Digital (115, 238, 120)-net over F₃, using

t-expansion [i] based on digital (113, 238, 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]

(115, 115+123, 782)-Net in Base 3 — Upper bound on s

There is no (115, 238, 783)-net in base 3, because

1 times m-reduction [i] would yield (115, 237, 783)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 124079 424585 800698 821209 643496 046003 896095 712041 590568 600474 088769 372373 956326 122956 278517 185698 510030 726983 086887 > 3²³⁷ [i]