MinT - Best Known (187−69, 187, s)-Nets in Base 3

Best Known (187−69, 187, s)-Nets in Base 3

(187−69, 187, 156)-Net over F₃ — Constructive and digital

Digital (118, 187, 156)-net over F₃, using

5 times m-reduction [i] based on digital (118, 192, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 96, 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]

(187−69, 187, 222)-Net over F₃ — Digital

Digital (118, 187, 222)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(187−69, 187, 2724)-Net in Base 3 — Upper bound on s

There is no (118, 187, 2725)-net in base 3, because

1 times m-reduction [i] would yield (118, 186, 2725)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 55825 567789 552733 644462 481142 216715 724574 234462 830297 201017 803384 913560 440872 816420 279569 > 3¹⁸⁶ [i]