MinT - Best Known (169−59, 169, s)-Nets in Base 3

Best Known (169−59, 169, s)-Nets in Base 3

(169−59, 169, 156)-Net over F₃ — Constructive and digital

Digital (110, 169, 156)-net over F₃, using

7 times m-reduction [i] based on digital (110, 176, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 88, 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]

(169−59, 169, 240)-Net over F₃ — Digital

Digital (110, 169, 240)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(169−59, 169, 3360)-Net in Base 3 — Upper bound on s

There is no (110, 169, 3361)-net in base 3, because

1 times m-reduction [i] would yield (110, 168, 3361)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 143 513953 071986 596319 121084 073679 083751 020989 036220 181757 601064 154239 198296 536531 > 3¹⁶⁸ [i]