MinT - Best Known (188−68, 188, s)-Nets in Base 3

Best Known (188−68, 188, s)-Nets in Base 3

(188−68, 188, 156)-Net over F₃ — Constructive and digital

Digital (120, 188, 156)-net over F₃, using

8 times m-reduction [i] based on digital (120, 196, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 98, 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]

(188−68, 188, 236)-Net over F₃ — Digital

Digital (120, 188, 236)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(188−68, 188, 2908)-Net in Base 3 — Upper bound on s

There is no (120, 188, 2909)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 501673 007622 822394 708758 560409 566019 796778 047390 227261 573251 884819 189431 739318 924474 616385 > 3¹⁸⁸ [i]