MinT - Best Known (205−113, 205, s)-Nets in Base 3

Best Known (205−113, 205, s)-Nets in Base 3

(205−113, 205, 64)-Net over F₃ — Constructive and digital

Digital (92, 205, 64)-net over F₃, using

t-expansion [i] based on digital (89, 205, 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]

(205−113, 205, 96)-Net over F₃ — Digital

Digital (92, 205, 96)-net over F₃, using

t-expansion [i] based on digital (89, 205, 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]

(205−113, 205, 540)-Net in Base 3 — Upper bound on s

There is no (92, 205, 541)-net in base 3, because

1 times m-reduction [i] would yield (92, 204, 541)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 22 702196 968239 735653 258229 286700 632279 225405 018463 828970 093709 724441 034071 407644 205757 918067 490977 > 3²⁰⁴ [i]