MinT - Best Known (188−67, 188, s)-Nets in Base 3

Best Known (188−67, 188, s)-Nets in Base 3

(188−67, 188, 156)-Net over F₃ — Constructive and digital

Digital (121, 188, 156)-net over F₃, using

10 times m-reduction [i] based on digital (121, 198, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 99, 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]

(188−67, 188, 247)-Net over F₃ — Digital

Digital (121, 188, 247)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−67, 188, 3294)-Net in Base 3 — Upper bound on s

There is no (121, 188, 3295)-net in base 3, because

1 times m-reduction [i] would yield (121, 187, 3295)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 167738 267550 582370 078302 928590 383018 353774 883103 519271 347701 895141 221400 294702 586027 252095 > 3¹⁸⁷ [i]