MinT - Best Known (158−87, 158, s)-Nets in Base 3

Best Known (158−87, 158, s)-Nets in Base 3

(158−87, 158, 52)-Net over F₃ — Constructive and digital

Digital (71, 158, 52)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (13, 56, 24)-net over F₃, using
  - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (15, 102, 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]

(158−87, 158, 84)-Net over F₃ — Digital

Digital (71, 158, 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]

(158−87, 158, 425)-Net in Base 3 — Upper bound on s

There is no (71, 158, 426)-net in base 3, because

1 times m-reduction [i] would yield (71, 157, 426)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 875 548729 866965 060198 033281 025770 194865 300152 914823 723701 943677 653362 023073 > 3¹⁵⁷ [i]