MinT - Best Known (159−71, 159, s)-Nets in Base 3

Best Known (159−71, 159, s)-Nets in Base 3

(159−71, 159, 80)-Net over F₃ — Constructive and digital

Digital (88, 159, 80)-net over F₃, using

1 times m-reduction [i] based on digital (88, 160, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 80, 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]

(159−71, 159, 113)-Net over F₃ — Digital

Digital (88, 159, 113)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(159−71, 159, 957)-Net in Base 3 — Upper bound on s

There is no (88, 159, 958)-net in base 3, because

1 times m-reduction [i] would yield (88, 158, 958)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2511 044135 901413 538790 176471 925162 651357 036421 677300 279615 857521 408485 135809 > 3¹⁵⁸ [i]