MinT - Best Known (237−143, 237, s)-Nets in Base 3

Best Known (237−143, 237, s)-Nets in Base 3

(237−143, 237, 64)-Net over F₃ — Constructive and digital

Digital (94, 237, 64)-net over F₃, using

t-expansion [i] based on digital (89, 237, 64)-net over F₃, using
- net from sequence [i] based on digital (89, 63)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(237−143, 237, 96)-Net over F₃ — Digital

Digital (94, 237, 96)-net over F₃, using

t-expansion [i] based on digital (89, 237, 96)-net over F₃, using
- net from sequence [i] based on digital (89, 95)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 89 and N(F) ≥ 96, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(237−143, 237, 458)-Net in Base 3 — Upper bound on s

There is no (94, 237, 459)-net in base 3, because

1 times m-reduction [i] would yield (94, 236, 459)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 42024 800465 410513 907721 084387 372485 448551 368732 220466 762424 267678 022726 878089 945252 467731 122098 802579 619435 580451 > 3²³⁶ [i]