MinT - Best Known (67−31, 67, s)-Nets in Base 4

Best Known (67−31, 67, s)-Nets in Base 4

(67−31, 67, 66)-Net over F₄ — Constructive and digital

Digital (36, 67, 66)-net over F₄, using

1 times m-reduction [i] based on digital (36, 68, 66)-net over F₄, using
- trace code for nets [i] based on digital (2, 34, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]

(67−31, 67, 76)-Net over F₄ — Digital

Digital (36, 67, 76)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−31, 67, 942)-Net in Base 4 — Upper bound on s

There is no (36, 67, 943)-net in base 4, because

1 times m-reduction [i] would yield (36, 66, 943)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5495 057527 707623 055587 722494 719550 853132 > 4⁶⁶ [i]