MinT - Best Known (164−75, 164, s)-Nets in Base 3

Best Known (164−75, 164, s)-Nets in Base 3

(164−75, 164, 72)-Net over F₃ — Constructive and digital

Digital (89, 164, 72)-net over F₃, using

trace code for nets [i] based on digital (7, 82, 36)-net over F₉, using
- net from sequence [i] based on digital (7, 35)-sequence over F₉, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 7 and N(F) ≥ 36, using
    - a function field by SÃ©mirat [i]

(164−75, 164, 108)-Net over F₃ — Digital

Digital (89, 164, 108)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(164−75, 164, 890)-Net in Base 3 — Upper bound on s

There is no (89, 164, 891)-net in base 3, because

1 times m-reduction [i] would yield (89, 163, 891)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 602295 330208 819823 575124 536478 874456 106893 382011 012599 391891 822787 959071 081839 > 3¹⁶³ [i]