MinT - Best Known (219−55, 219, s)-Nets in Base 3

Best Known (219−55, 219, s)-Nets in Base 3

(219−55, 219, 324)-Net over F₃ — Constructive and digital

Digital (164, 219, 324)-net over F₃, using

trace code for nets [i] based on digital (18, 73, 108)-net over F₂₇, using
- net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
    - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(219−55, 219, 899)-Net over F₃ — Digital

Digital (164, 219, 899)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(219−55, 219, 38850)-Net in Base 3 — Upper bound on s

There is no (164, 219, 38851)-net in base 3, because

1 times m-reduction [i] would yield (164, 218, 38851)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 102 962653 477157 991790 949978 574484 855235 805270 063654 149075 417588 765851 730315 104055 005869 471086 710772 326139 > 3²¹⁸ [i]