MinT - Best Known (223−81, 223, s)-Nets in Base 3

Best Known (223−81, 223, s)-Nets in Base 3

(223−81, 223, 156)-Net over F₃ — Constructive and digital

Digital (142, 223, 156)-net over F₃, using

17 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]

(223−81, 223, 271)-Net over F₃ — Digital

Digital (142, 223, 271)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(223−81, 223, 3466)-Net in Base 3 — Upper bound on s

There is no (142, 223, 3467)-net in base 3, because

1 times m-reduction [i] would yield (142, 222, 3467)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8386 749512 677639 746975 836128 517686 509119 117302 897898 599716 074744 659731 548110 710787 774736 592280 035387 877425 > 3²²² [i]