MinT - Best Known (250−165, 250, s)-Nets in Base 3

Best Known (250−165, 250, s)-Nets in Base 3

(250−165, 250, 60)-Net over F₃ — Constructive and digital

Digital (85, 250, 60)-net over F₃, using

net from sequence [i] based on digital (85, 59)-sequence over F₃, using
- base reduction for sequences [i] based on digital (13, 59)-sequence over F₉, using
  - s-reduction 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]

(250−165, 250, 84)-Net over F₃ — Digital

Digital (85, 250, 84)-net over F₃, using

t-expansion [i] based on digital (71, 250, 84)-net over F₃, using
- net from sequence [i] based on digital (71, 83)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(250−165, 250, 269)-Net over F₃ — Upper bound on s (digital)

There is no digital (85, 250, 270)-net over F₃, because

extracting embedded orthogonal array [i] would yield linear OA(3²⁵⁰, 270, F₃, 165) (dual of [270, 20, 166]-code), but
- residual code [i] would yield OA(3⁸⁵, 104, S₃, 55), but
  - the linear programming bound shows that MÂ â‰¥ 4744 075529 079928 845960 058598 807593 969359 823300 884489 / 94629 133984 > 3⁸⁵ [i]

(250−165, 250, 364)-Net in Base 3 — Upper bound on s

There is no (85, 250, 365)-net in base 3, because

1 times m-reduction [i] would yield (85, 249, 365)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 68344 153945 943613 250955 700416 657173 979643 709508 241423 765516 785097 190285 375609 796823 822011 370533 535749 280524 750910 377377 > 3²⁴⁹ [i]