MinT - Best Known (72−33, 72, s)-Nets in Base 4

Best Known (72−33, 72, s)-Nets in Base 4

(72−33, 72, 76)-Net over F₄ — Constructive and digital

Digital (39, 72, 76)-net over F₄, using

trace code for nets [i] based on digital (3, 36, 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]

(72−33, 72, 82)-Net over F₄ — Digital

Digital (39, 72, 82)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(72−33, 72, 1051)-Net in Base 4 — Upper bound on s

There is no (39, 72, 1052)-net in base 4, because

1 times m-reduction [i] would yield (39, 71, 1052)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 616128 788626 935257 638015 180365 243798 066503 > 4⁷¹ [i]