MinT - Best Known (80−31, 80, s)-Nets in Base 3

Best Known (80−31, 80, s)-Nets in Base 3

(80−31, 80, 80)-Net over F₃ — Constructive and digital

Digital (49, 80, 80)-net over F₃, using

2 times m-reduction [i] based on digital (49, 82, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 41, 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]

(80−31, 80, 98)-Net over F₃ — Digital

Digital (49, 80, 98)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(80−31, 80, 1031)-Net in Base 3 — Upper bound on s

There is no (49, 80, 1032)-net in base 3, because

1 times m-reduction [i] would yield (49, 79, 1032)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 49 490134 288907 666329 195822 806305 590881 > 3⁷⁹ [i]