MinT - Best Known (89, 89+71, s)-Nets in Base 3

Best Known (89, 89+71, s)-Nets in Base 3

(89, 89+71, 80)-Net over F₃ — Constructive and digital

Digital (89, 160, 80)-net over F₃, using

2 times m-reduction [i] based on digital (89, 162, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 81, 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]

(89, 89+71, 116)-Net over F₃ — Digital

Digital (89, 160, 116)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+71, 988)-Net in Base 3 — Upper bound on s

There is no (89, 160, 989)-net in base 3, because

1 times m-reduction [i] would yield (89, 159, 989)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 7375 754744 277575 335954 252651 232494 508648 489528 869266 061032 229540 732847 324427 > 3¹⁵⁹ [i]