MinT - Best Known (117, 117+117, s)-Nets in Base 3

Best Known (117, 117+117, s)-Nets in Base 3

(117, 117+117, 75)-Net over F₃ — Constructive and digital

Digital (117, 234, 75)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (27, 85, 37)-net over F₃, using
  - net from sequence [i] based on digital (27, 36)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₃ with g(F)Â =Â 26, N(F)Â =Â 36, and 1 place with degree 2 [i] based on function field F/F₃ with g(F) = 26 and N(F) ≥ 36, using an explicitly constructive algebraic function field [i]
2. digital (32, 149, 38)-net over F₃, using
  - net from sequence [i] based on digital (32, 37)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 32 and N(F) ≥ 38, using
      - an explicitly constructive algebraic function field [i]

(117, 117+117, 120)-Net over F₃ — Digital

Digital (117, 234, 120)-net over F₃, using

t-expansion [i] based on digital (113, 234, 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]

(117, 117+117, 870)-Net in Base 3 — Upper bound on s

There is no (117, 234, 871)-net in base 3, because

1 times m-reduction [i] would yield (117, 233, 871)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1516 651218 260715 018864 847719 686363 226919 634535 859796 308534 394272 439739 551786 806851 568431 098525 860840 696931 182557 > 3²³³ [i]