MinT - Best Known (215−119, 215, s)-Nets in Base 3

Best Known (215−119, 215, s)-Nets in Base 3

(215−119, 215, 64)-Net over F₃ — Constructive and digital

Digital (96, 215, 64)-net over F₃, using

t-expansion [i] based on digital (89, 215, 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]

(215−119, 215, 96)-Net over F₃ — Digital

Digital (96, 215, 96)-net over F₃, using

t-expansion [i] based on digital (89, 215, 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]

(215−119, 215, 557)-Net in Base 3 — Upper bound on s

There is no (96, 215, 558)-net in base 3, because

1 times m-reduction [i] would yield (96, 214, 558)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 358034 202058 040784 261811 643406 810913 209741 081427 603349 297093 482331 887014 261889 701060 626499 859776 502545 > 3²¹⁴ [i]