MinT - Best Known (250−135, 250, s)-Nets in Base 3

Best Known (250−135, 250, s)-Nets in Base 3

(250−135, 250, 74)-Net over F₃ — Constructive and digital

Digital (115, 250, 74)-net over F₃, using

t-expansion [i] based on digital (107, 250, 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]

(250−135, 250, 120)-Net over F₃ — Digital

Digital (115, 250, 120)-net over F₃, using

t-expansion [i] based on digital (113, 250, 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]

(250−135, 250, 700)-Net in Base 3 — Upper bound on s

There is no (115, 250, 701)-net in base 3, because

1 times m-reduction [i] would yield (115, 249, 701)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 66075 876434 576213 947257 922098 889242 829643 317956 333474 547259 460947 820125 746208 059433 375177 703504 018208 109471 567473 262091 > 3²⁴⁹ [i]