MinT - Best Known (243−129, 243, s)-Nets in Base 3

Best Known (243−129, 243, s)-Nets in Base 3

(243−129, 243, 74)-Net over F₃ — Constructive and digital

Digital (114, 243, 74)-net over F₃, using

t-expansion [i] based on digital (107, 243, 74)-net over F₃, using
- net from sequence [i] based on digital (107, 73)-sequence over F₃, using
  - base reduction for sequences [i] based on digital (17, 73)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 17 and N(F) ≥ 74, using
      - a function field by GarcÃa and Quoos [i]

(243−129, 243, 120)-Net over F₃ — Digital

Digital (114, 243, 120)-net over F₃, using

t-expansion [i] based on digital (113, 243, 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]

(243−129, 243, 724)-Net in Base 3 — Upper bound on s

There is no (114, 243, 725)-net in base 3, because

1 times m-reduction [i] would yield (114, 242, 725)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 30 591335 210956 884717 838779 075859 009373 236200 368525 591696 294966 148001 362090 298878 878125 201047 706662 314967 897554 542337 > 3²⁴² [i]