MinT - Best Known (85, 85+65, s)-Nets in Base 3

Best Known (85, 85+65, s)-Nets in Base 3

(85, 85+65, 80)-Net over F₃ — Constructive and digital

Digital (85, 150, 80)-net over F₃, using

4 times m-reduction [i] based on digital (85, 154, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 77, 40)-net over F₉, using
  - net from sequence [i] based on digital (8, 39)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 8 and N(F) ≥ 40, using
      - a function field by SÃ©mirat [i]

(85, 85+65, 117)-Net over F₃ — Digital

Digital (85, 150, 117)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(85, 85+65, 1034)-Net in Base 3 — Upper bound on s

There is no (85, 150, 1035)-net in base 3, because

1 times m-reduction [i] would yield (85, 149, 1035)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 126633 848632 401060 141355 291959 814280 897980 730533 542408 278575 399372 407745 > 3¹⁴⁹ [i]