MinT - Best Known (177−57, 177, s)-Nets in Base 3

Best Known (177−57, 177, s)-Nets in Base 3

(177−57, 177, 156)-Net over F₃ — Constructive and digital

Digital (120, 177, 156)-net over F₃, using

19 times m-reduction [i] based on digital (120, 196, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 98, 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]

(177−57, 177, 320)-Net over F₃ — Digital

Digital (120, 177, 320)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(177−57, 177, 5609)-Net in Base 3 — Upper bound on s

There is no (120, 177, 5610)-net in base 3, because

1 times m-reduction [i] would yield (120, 176, 5610)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 944065 152278 243337 393816 800584 967446 068561 609355 566037 434727 583366 454345 777418 025241 > 3¹⁷⁶ [i]