MinT - Best Known (144, 231, s)-Nets in Base 3

Best Known (144, 231, s)-Nets in Base 3

(144, 231, 156)-Net over F₃ — Constructive and digital

Digital (144, 231, 156)-net over F₃, using

13 times m-reduction [i] based on digital (144, 244, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 122, 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]

(144, 231, 252)-Net over F₃ — Digital

Digital (144, 231, 252)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(144, 231, 2967)-Net in Base 3 — Upper bound on s

There is no (144, 231, 2968)-net in base 3, because

1 times m-reduction [i] would yield (144, 230, 2968)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55 342970 082494 255149 376813 566969 833493 408569 892166 470448 627431 335364 369205 440234 685990 811033 535108 249142 242785 > 3²³⁰ [i]