MinT - Best Known (89−33, 89, s)-Nets in Base 3

Best Known (89−33, 89, s)-Nets in Base 3

(89−33, 89, 80)-Net over F₃ — Constructive and digital

Digital (56, 89, 80)-net over F₃, using

7 times m-reduction [i] based on digital (56, 96, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 48, 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]

(89−33, 89, 121)-Net over F₃ — Digital

Digital (56, 89, 121)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−33, 89, 1415)-Net in Base 3 — Upper bound on s

There is no (56, 89, 1416)-net in base 3, because

1 times m-reduction [i] would yield (56, 88, 1416)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 973156 540137 782470 206382 848072 392225 137665 > 3⁸⁸ [i]