MinT - Best Known (233−91, 233, s)-Nets in Base 3

Best Known (233−91, 233, s)-Nets in Base 3

(233−91, 233, 156)-Net over F₃ — Constructive and digital

Digital (142, 233, 156)-net over F₃, using

7 times m-reduction [i] based on digital (142, 240, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 120, 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]

(233−91, 233, 228)-Net over F₃ — Digital

Digital (142, 233, 228)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(233−91, 233, 2496)-Net in Base 3 — Upper bound on s

There is no (142, 233, 2497)-net in base 3, because

1 times m-reduction [i] would yield (142, 232, 2497)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 495 582337 868283 226334 227896 267969 298032 760132 577790 013828 278495 329862 612878 597939 648202 979797 686921 136491 886291 > 3²³² [i]