MinT - Best Known (188−70, 188, s)-Nets in Base 3

Best Known (188−70, 188, s)-Nets in Base 3

(188−70, 188, 156)-Net over F₃ — Constructive and digital

Digital (118, 188, 156)-net over F₃, using

4 times m-reduction [i] based on digital (118, 192, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 96, 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]

(188−70, 188, 217)-Net over F₃ — Digital

Digital (118, 188, 217)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−70, 188, 2506)-Net in Base 3 — Upper bound on s

There is no (118, 188, 2507)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 499815 667632 234531 647209 221947 484616 165718 029706 598190 808495 023755 831097 076551 918009 937163 > 3¹⁸⁸ [i]