MinT - Best Known (87−29, 87, s)-Nets in Base 3

Best Known (87−29, 87, s)-Nets in Base 3

(87−29, 87, 128)-Net over F₃ — Constructive and digital

Digital (58, 87, 128)-net over F₃, using

3 times m-reduction [i] based on digital (58, 90, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 45, 64)-net over F₉, using
  - net from sequence [i] 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]

(87−29, 87, 162)-Net over F₃ — Digital

Digital (58, 87, 162)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−29, 87, 2564)-Net in Base 3 — Upper bound on s

There is no (58, 87, 2565)-net in base 3, because

1 times m-reduction [i] would yield (58, 86, 2565)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 107915 004988 482306 830232 380970 250590 639457 > 3⁸⁶ [i]