MinT - Best Known (109−45, 109, s)-Nets in Base 3

Best Known (109−45, 109, s)-Nets in Base 3

(109−45, 109, 80)-Net over F₃ — Constructive and digital

Digital (64, 109, 80)-net over F₃, using

3 times m-reduction [i] based on digital (64, 112, 80)-net over F₃, using
- trace code for nets [i] based on digital (8, 56, 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]

(109−45, 109, 103)-Net over F₃ — Digital

Digital (64, 109, 103)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−45, 109, 974)-Net in Base 3 — Upper bound on s

There is no (64, 109, 975)-net in base 3, because

1 times m-reduction [i] would yield (64, 108, 975)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3439 970457 998530 742450 865230 341112 344447 054959 740661 > 3¹⁰⁸ [i]