MinT - Best Known (178−64, 178, s)-Nets in Base 3

Best Known (178−64, 178, s)-Nets in Base 3

(178−64, 178, 156)-Net over F₃ — Constructive and digital

Digital (114, 178, 156)-net over F₃, using

6 times m-reduction [i] based on digital (114, 184, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 92, 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]

(178−64, 178, 229)-Net over F₃ — Digital

Digital (114, 178, 229)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(178−64, 178, 2851)-Net in Base 3 — Upper bound on s

There is no (114, 178, 2852)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 8 502328 643358 446927 362108 034890 696498 940416 848786 273294 811380 222057 002281 011555 675393 > 3¹⁷⁸ [i]