MinT - Best Known (123, 123+119, s)-Nets in Base 3

Best Known (123, 123+119, s)-Nets in Base 3

(123, 123+119, 78)-Net over F₃ — Constructive and digital

Digital (123, 242, 78)-net over F₃, using

t-expansion [i] based on digital (121, 242, 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]

(123, 123+119, 124)-Net over F₃ — Digital

Digital (123, 242, 124)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+119, 957)-Net in Base 3 — Upper bound on s

There is no (123, 242, 958)-net in base 3, because

1 times m-reduction [i] would yield (123, 241, 958)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 10 086059 688469 981238 122248 740165 158179 290730 180930 402425 583353 211209 670319 163685 223346 631730 519041 345097 386381 646545 > 3²⁴¹ [i]