MinT - Best Known (68, 68+47, s)-Nets in Base 3

Best Known (68, 68+47, s)-Nets in Base 3

(68, 68+47, 80)-Net over F₃ — Constructive and digital

Digital (68, 115, 80)-net over F₃, using

5 times m-reduction [i] based on digital (68, 120, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 60, 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]

(68, 68+47, 111)-Net over F₃ — Digital

Digital (68, 115, 111)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+47, 1069)-Net in Base 3 — Upper bound on s

There is no (68, 115, 1070)-net in base 3, because

1 times m-reduction [i] would yield (68, 114, 1070)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2 472498 894566 891512 170672 406557 439636 565380 584787 683033 > 3¹¹⁴ [i]