MinT - Best Known (164−93, 164, s)-Nets in Base 3

Best Known (164−93, 164, s)-Nets in Base 3

(164−93, 164, 48)-Net over F₃ — Constructive and digital

Digital (71, 164, 48)-net over F₃, using

t-expansion [i] based on digital (45, 164, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(164−93, 164, 84)-Net over F₃ — Digital

Digital (71, 164, 84)-net over F₃, using

net from sequence [i] based on digital (71, 83)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 71 and N(F) ≥ 84, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(164−93, 164, 397)-Net in Base 3 — Upper bound on s

There is no (71, 164, 398)-net in base 3, because

1 times m-reduction [i] would yield (71, 163, 398)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 603061 858085 109019 520165 549067 679546 486518 190322 078557 820843 652624 241083 638741 > 3¹⁶³ [i]