MinT - Best Known (129−57, 129, s)-Nets in Base 3

Best Known (129−57, 129, s)-Nets in Base 3

(129−57, 129, 72)-Net over F₃ — Constructive and digital

Digital (72, 129, 72)-net over F₃, using

1 times m-reduction [i] based on digital (72, 130, 72)-net over F₃, using
- trace code for nets [i] based on digital (7, 65, 36)-net over F₉, using
  - net from sequence [i] based on digital (7, 35)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 7 and N(F) ≥ 36, using
      - a function field by SÃ©mirat [i]

(129−57, 129, 98)-Net over F₃ — Digital

Digital (72, 129, 98)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(129−57, 129, 830)-Net in Base 3 — Upper bound on s

There is no (72, 129, 831)-net in base 3, because

1 times m-reduction [i] would yield (72, 128, 831)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 12 142275 025307 033315 939458 065484 971514 769984 822743 173894 581193 > 3¹²⁸ [i]