MinT - Best Known (109−31, 109, s)-Nets in Base 3

Best Known (109−31, 109, s)-Nets in Base 3

(109−31, 109, 204)-Net over F₃ — Constructive and digital

Digital (78, 109, 204)-net over F₃, using

3¹ times duplication [i] based on digital (77, 108, 204)-net over F₃, using
- trace code for nets [i] based on digital (5, 36, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]

(109−31, 109, 326)-Net over F₃ — Digital

Digital (78, 109, 326)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−31, 109, 8735)-Net in Base 3 — Upper bound on s

There is no (78, 109, 8736)-net in base 3, because

1 times m-reduction [i] would yield (78, 108, 8736)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 3384 043006 909505 231471 314074 573321 950044 299705 491841 > 3¹⁰⁸ [i]