MinT - Best Known (68−29, 68, s)-Nets in Base 4

Best Known (68−29, 68, s)-Nets in Base 4

(68−29, 68, 98)-Net over F₄ — Constructive and digital

Digital (39, 68, 98)-net over F₄, using

trace code for nets [i] based on digital (5, 34, 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]

(68−29, 68, 100)-Net over F₄ — Digital

Digital (39, 68, 100)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(68−29, 68, 1522)-Net in Base 4 — Upper bound on s

There is no (39, 68, 1523)-net in base 4, because

1 times m-reduction [i] would yield (39, 67, 1523)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 21947 233714 848490 409596 655942 635898 586076 > 4⁶⁷ [i]