MinT - Best Known (243−123, 243, s)-Nets in Base 3

Best Known (243−123, 243, s)-Nets in Base 3

(243−123, 243, 77)-Net over F₃ — Constructive and digital

Digital (120, 243, 77)-net over F₃, using

net from sequence [i] based on digital (120, 76)-sequence over F₃, using
- base reduction for sequences [i] based on digital (22, 76)-sequence over F₉, using
  - s-reduction based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(243−123, 243, 120)-Net over F₃ — Digital

Digital (120, 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]

(243−123, 243, 861)-Net in Base 3 — Upper bound on s

There is no (120, 243, 862)-net in base 3, because

1 times m-reduction [i] would yield (120, 242, 862)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 29 806899 018755 886862 999945 556423 356743 523891 002536 246177 685990 978482 018366 690930 666652 311099 116665 134907 903970 173285 > 3²⁴² [i]