MinT - Best Known (34−29, 34, s)-Nets in Base 64

Best Known (34−29, 34, s)-Nets in Base 64

(34−29, 34, 128)-Net over F₆₄ — Constructive and digital

Digital (5, 34, 128)-net over F₆₄, using

net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
  - a function field by SÃ©mirat [i]

(34−29, 34, 129)-Net in Base 64 — Constructive

(5, 34, 129)-net in base 64, using

1 times m-reduction [i] based on (5, 35, 129)-net in base 64, using
- base change [i] based on digital (0, 30, 129)-net over F₁₂₈, using
  - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
      - the rational function field F₁₂₈(x) [i]
    - Niederreiter sequence [i]

(34−29, 34, 133)-Net over F₆₄ — Digital

Digital (5, 34, 133)-net over F₆₄, using

net from sequence [i] based on digital (5, 132)-sequence over F₆₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 133, using
  - a curve from the manYPoints database [i]

(34−29, 34, 1729)-Net in Base 64 — Upper bound on s

There is no (5, 34, 1730)-net in base 64, because

1 times m-reduction [i] would yield (5, 33, 1730)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 404170 926022 668972 720944 127498 836359 364827 548362 986362 738976 > 64³³ [i]