MinT - Best Known (218−105, 218, s)-Nets in Base 3

Best Known (218−105, 218, s)-Nets in Base 3

(218−105, 218, 76)-Net over F₃ — Constructive and digital

Digital (113, 218, 76)-net over F₃, using

1 times m-reduction [i] based on digital (113, 219, 76)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (15, 68, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]
  2. digital (45, 151, 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]

(218−105, 218, 120)-Net over F₃ — Digital

Digital (113, 218, 120)-net over F₃, using

net from sequence [i] based on digital (113, 119)-sequence over F₃, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 113 and N(F) ≥ 120, using
  - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(218−105, 218, 940)-Net in Base 3 — Upper bound on s

There is no (113, 218, 941)-net in base 3, because

1 times m-reduction [i] would yield (113, 217, 941)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 35 661949 922759 983569 294394 427440 896096 752228 733032 423120 123360 453549 080940 139525 066540 256710 277508 368593 > 3²¹⁷ [i]