MinT - Best Known (166−67, 166, s)-Nets in Base 3

Best Known (166−67, 166, s)-Nets in Base 3

(166−67, 166, 128)-Net over F₃ — Constructive and digital

Digital (99, 166, 128)-net over F₃, using

6 times m-reduction [i] based on digital (99, 172, 128)-net over F₃, using
- trace code for nets [i] based on digital (13, 86, 64)-net over F₉, using
  - net from sequence [i] based on digital (13, 63)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 13 and N(F) ≥ 64, using
      - a function field by GarcÃa and Quoos [i]

(166−67, 166, 155)-Net over F₃ — Digital

Digital (99, 166, 155)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−67, 166, 1567)-Net in Base 3 — Upper bound on s

There is no (99, 166, 1568)-net in base 3, because

1 times m-reduction [i] would yield (99, 165, 1568)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 411277 554922 933203 826043 518409 052866 912037 775238 258225 009963 273808 068475 634753 > 3¹⁶⁵ [i]