MinT - Best Known (250−129, 250, s)-Nets in Base 3

Best Known (250−129, 250, s)-Nets in Base 3

(250−129, 250, 78)-Net over F₃ — Constructive and digital

Digital (121, 250, 78)-net over F₃, using

net from sequence [i] based on digital (121, 77)-sequence over F₃, using
- base reduction for sequences [i] 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]

(250−129, 250, 120)-Net over F₃ — Digital

Digital (121, 250, 120)-net over F₃, using

t-expansion [i] based on digital (113, 250, 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]

(250−129, 250, 824)-Net in Base 3 — Upper bound on s

There is no (121, 250, 825)-net in base 3, because

1 times m-reduction [i] would yield (121, 249, 825)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 65989 887198 067137 174012 252524 227780 180651 487406 385304 664190 676735 617251 539254 144279 668170 108297 878059 337438 552004 377857 > 3²⁴⁹ [i]