MinT - Best Known (232−115, 232, s)-Nets in Base 3

Best Known (232−115, 232, s)-Nets in Base 3

(232−115, 232, 76)-Net over F₃ — Constructive and digital

Digital (117, 232, 76)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 72, 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, 160, 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]

(232−115, 232, 120)-Net over F₃ — Digital

Digital (117, 232, 120)-net over F₃, using

t-expansion [i] based on digital (113, 232, 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]

(232−115, 232, 892)-Net in Base 3 — Upper bound on s

There is no (117, 232, 893)-net in base 3, because

1 times m-reduction [i] would yield (117, 231, 893)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 170 891497 444195 641987 150615 175551 192217 426526 772212 640175 742374 538342 778776 775215 208658 954758 295896 146741 368347 > 3²³¹ [i]