MinT - Best Known (89−41, 89, s)-Nets in Base 3

Best Known (89−41, 89, s)-Nets in Base 3

(89−41, 89, 56)-Net over F₃ — Constructive and digital

Digital (48, 89, 56)-net over F₃, using

1 times m-reduction [i] based on digital (48, 90, 56)-net over F₃, using
- trace code for nets [i] based on digital (3, 45, 28)-net over F₉, using
  - net from sequence [i] based on digital (3, 27)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 3 and N(F) ≥ 28, using
      - the Hermitian function field over F₉ [i]

(89−41, 89, 65)-Net over F₃ — Digital

Digital (48, 89, 65)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89−41, 89, 502)-Net in Base 3 — Upper bound on s

There is no (48, 89, 503)-net in base 3, because

1 times m-reduction [i] would yield (48, 88, 503)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 980020 005859 594115 026102 716437 990706 119001 > 3⁸⁸ [i]