MinT - Best Known (187−70, 187, s)-Nets in Base 3

Best Known (187−70, 187, s)-Nets in Base 3

(187−70, 187, 156)-Net over F₃ — Constructive and digital

Digital (117, 187, 156)-net over F₃, using

3 times m-reduction [i] based on digital (117, 190, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 95, 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]

(187−70, 187, 213)-Net over F₃ — Digital

Digital (117, 187, 213)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−70, 187, 2428)-Net in Base 3 — Upper bound on s

There is no (117, 187, 2429)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 167849 978688 651568 726657 665953 769946 517244 612324 194518 416884 245902 308411 938396 442604 069003 > 3¹⁸⁷ [i]