MinT - Best Known (68−27, 68, s)-Nets in Base 5

Best Known (68−27, 68, s)-Nets in Base 5

(68−27, 68, 132)-Net over F₅ — Constructive and digital

Digital (41, 68, 132)-net over F₅, using

6 times m-reduction [i] based on digital (41, 74, 132)-net over F₅, using
- trace code for nets [i] based on digital (4, 37, 66)-net over F₂₅, using
  - net from sequence [i] based on digital (4, 65)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 4 and N(F) ≥ 66, using
      - a function field by GarcÃa and Quoos [i]

(68−27, 68, 176)-Net over F₅ — Digital

Digital (41, 68, 176)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−27, 68, 5662)-Net in Base 5 — Upper bound on s

There is no (41, 68, 5663)-net in base 5, because

1 times m-reduction [i] would yield (41, 67, 5663)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 67813 131320 044788 378648 472281 649591 692982 386765 > 5⁶⁷ [i]