MinT - Best Known (236−89, 236, s)-Nets in Base 3

Best Known (236−89, 236, s)-Nets in Base 3

(236−89, 236, 156)-Net over F₃ — Constructive and digital

Digital (147, 236, 156)-net over F₃, using

14 times m-reduction [i] based on digital (147, 250, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 125, 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]

(236−89, 236, 256)-Net over F₃ — Digital

Digital (147, 236, 256)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(236−89, 236, 3005)-Net in Base 3 — Upper bound on s

There is no (147, 236, 3006)-net in base 3, because

1 times m-reduction [i] would yield (147, 235, 3006)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 13340 433165 094927 446469 702548 139317 739104 607479 697510 040770 827158 308314 489026 765591 789366 206397 151343 070149 027353 > 3²³⁵ [i]