MinT - Best Known (89, 89+68, s)-Nets in Base 3

Best Known (89, 89+68, s)-Nets in Base 3

(89, 89+68, 80)-Net over F₃ — Constructive and digital

Digital (89, 157, 80)-net over F₃, using

5 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+68, 122)-Net over F₃ — Digital

Digital (89, 157, 122)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 89+68, 1047)-Net in Base 3 — Upper bound on s

There is no (89, 157, 1048)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 824 016887 798862 974516 193093 510155 061013 367788 096712 816748 020036 164676 240145 > 3¹⁵⁷ [i]