MinT - Best Known (217−77, 217, s)-Nets in Base 3

Best Known (217−77, 217, s)-Nets in Base 3

(217−77, 217, 156)-Net over F₃ — Constructive and digital

Digital (140, 217, 156)-net over F₃, using

19 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]

(217−77, 217, 283)-Net over F₃ — Digital

Digital (140, 217, 283)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(217−77, 217, 3833)-Net in Base 3 — Upper bound on s

There is no (140, 217, 3834)-net in base 3, because

1 times m-reduction [i] would yield (140, 216, 3834)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 11 465372 705005 607936 285689 145513 435016 931569 341067 509384 284836 354992 995075 476730 067217 505573 271493 482445 > 3²¹⁶ [i]