MinT - Best Known (166−61, 166, s)-Nets in Base 3

Best Known (166−61, 166, s)-Nets in Base 3

(166−61, 166, 156)-Net over F₃ — Constructive and digital

Digital (105, 166, 156)-net over F₃, using

trace code for nets [i] based on digital (22, 83, 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]

(166−61, 166, 203)-Net over F₃ — Digital

Digital (105, 166, 203)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(166−61, 166, 2505)-Net in Base 3 — Upper bound on s

There is no (105, 166, 2506)-net in base 3, because

1 times m-reduction [i] would yield (105, 165, 2506)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 5 350765 088404 229518 915774 924942 766656 099974 463285 808017 214779 319584 778539 258205 > 3¹⁶⁵ [i]