MinT - Best Known (69−31, 69, s)-Nets in Base 4

Best Known (69−31, 69, s)-Nets in Base 4

(69−31, 69, 76)-Net over F₄ — Constructive and digital

Digital (38, 69, 76)-net over F₄, using

1 times m-reduction [i] based on digital (38, 70, 76)-net over F₄, using
- trace code for nets [i] based on digital (3, 35, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(69−31, 69, 85)-Net over F₄ — Digital

Digital (38, 69, 85)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−31, 69, 1136)-Net in Base 4 — Upper bound on s

There is no (38, 69, 1137)-net in base 4, because

1 times m-reduction [i] would yield (38, 68, 1137)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 88073 393416 624619 208768 914338 268560 000240 > 4⁶⁸ [i]