MinT - Best Known (244−123, 244, s)-Nets in Base 3

Best Known (244−123, 244, s)-Nets in Base 3

(244−123, 244, 78)-Net over F₃ — Constructive and digital

Digital (121, 244, 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]

(244−123, 244, 120)-Net over F₃ — Digital

Digital (121, 244, 120)-net over F₃, using

t-expansion [i] based on digital (113, 244, 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]

(244−123, 244, 878)-Net in Base 3 — Upper bound on s

There is no (121, 244, 879)-net in base 3, because

1 times m-reduction [i] would yield (121, 243, 879)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 91 137199 529779 840399 056212 071355 101104 216408 792168 685287 487484 638575 193680 009630 066433 518717 113260 867317 168010 457319 > 3²⁴³ [i]