MinT - Best Known (223−109, 223, s)-Nets in Base 3

Best Known (223−109, 223, s)-Nets in Base 3

(223−109, 223, 76)-Net over F₃ — Constructive and digital

Digital (114, 223, 76)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 69, 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, 154, 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]

(223−109, 223, 120)-Net over F₃ — Digital

Digital (114, 223, 120)-net over F₃, using

t-expansion [i] based on digital (113, 223, 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]

(223−109, 223, 907)-Net in Base 3 — Upper bound on s

There is no (114, 223, 908)-net in base 3, because

1 times m-reduction [i] would yield (114, 222, 908)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8752 512264 650991 118072 157629 937944 292885 706410 767776 816487 150053 024748 794643 681620 500755 282820 312143 453449 > 3²²² [i]