MinT - Best Known (122, 245, s)-Nets in Base 3

Best Known (122, 245, s)-Nets in Base 3

(122, 245, 78)-Net over F₃ — Constructive and digital

Digital (122, 245, 78)-net over F₃, using

t-expansion [i] based on digital (121, 245, 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]

(122, 245, 120)-Net over F₃ — Digital

Digital (122, 245, 120)-net over F₃, using

t-expansion [i] based on digital (113, 245, 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]

(122, 245, 895)-Net in Base 3 — Upper bound on s

There is no (122, 245, 896)-net in base 3, because

1 times m-reduction [i] would yield (122, 244, 896)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 273 095468 514132 134598 712185 626930 595896 191837 322606 604423 651676 610407 261077 930808 638158 689199 137648 754122 699673 444097 > 3²⁴⁴ [i]