MinT - Best Known (69−29, 69, s)-Nets in Base 4

Best Known (69−29, 69, s)-Nets in Base 4

(69−29, 69, 98)-Net over F₄ — Constructive and digital

Digital (40, 69, 98)-net over F₄, using

1 times m-reduction [i] based on digital (40, 70, 98)-net over F₄, using
- trace code for nets [i] based on digital (5, 35, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]

(69−29, 69, 106)-Net over F₄ — Digital

Digital (40, 69, 106)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−29, 69, 1681)-Net in Base 4 — Upper bound on s

There is no (40, 69, 1682)-net in base 4, because

1 times m-reduction [i] would yield (40, 68, 1682)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 87293 614664 360100 378893 308300 230860 201800 > 4⁶⁸ [i]