MinT - Best Known (190−71, 190, s)-Nets in Base 3

Best Known (190−71, 190, s)-Nets in Base 3

(190−71, 190, 156)-Net over F₃ — Constructive and digital

Digital (119, 190, 156)-net over F₃, using

4 times m-reduction [i] based on digital (119, 194, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 97, 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]

(190−71, 190, 217)-Net over F₃ — Digital

Digital (119, 190, 217)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(190−71, 190, 2588)-Net in Base 3 — Upper bound on s

There is no (119, 190, 2589)-net in base 3, because

1 times m-reduction [i] would yield (119, 189, 2589)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 519023 466190 811866 723402 626787 703447 166441 502094 826771 321488 330702 167625 913979 651508 877323 > 3¹⁸⁹ [i]