MinT - Best Known (231−91, 231, s)-Nets in Base 3

Best Known (231−91, 231, s)-Nets in Base 3

(231−91, 231, 156)-Net over F₃ — Constructive and digital

Digital (140, 231, 156)-net over F₃, using

5 times m-reduction [i] based on digital (140, 236, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 118, 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]

(231−91, 231, 221)-Net over F₃ — Digital

Digital (140, 231, 221)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(231−91, 231, 2375)-Net in Base 3 — Upper bound on s

There is no (140, 231, 2376)-net in base 3, because

1 times m-reduction [i] would yield (140, 230, 2376)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55 148922 755382 942189 047519 683331 996726 689454 474586 922731 668850 651741 407010 723444 033601 527765 381330 664573 158033 > 3²³⁰ [i]