MinT - Best Known (123, 123+77, s)-Nets in Base 3

Best Known (123, 123+77, s)-Nets in Base 3

(123, 123+77, 156)-Net over F₃ — Constructive and digital

Digital (123, 200, 156)-net over F₃, using

2 times m-reduction [i] based on digital (123, 202, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 101, 78)-net over F₉, using
  - net from sequence [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+77, 208)-Net over F₃ — Digital

Digital (123, 200, 208)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+77, 2330)-Net in Base 3 — Upper bound on s

There is no (123, 200, 2331)-net in base 3, because

1 times m-reduction [i] would yield (123, 199, 2331)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 88898 695907 033850 372860 366715 382577 347172 135630 894858 050248 253045 706609 785228 958840 529605 328333 > 3¹⁹⁹ [i]