MinT - Best Known (233−87, 233, s)-Nets in Base 3

Best Known (233−87, 233, s)-Nets in Base 3

(233−87, 233, 156)-Net over F₃ — Constructive and digital

Digital (146, 233, 156)-net over F₃, using

15 times m-reduction [i] based on digital (146, 248, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 124, 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]

(233−87, 233, 260)-Net over F₃ — Digital

Digital (146, 233, 260)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(233−87, 233, 3124)-Net in Base 3 — Upper bound on s

There is no (146, 233, 3125)-net in base 3, because

1 times m-reduction [i] would yield (146, 232, 3125)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 492 679668 426564 687589 345007 272663 804880 609748 811211 547435 003913 282195 896204 122160 620267 988478 894766 837389 570251 > 3²³² [i]