MinT - Best Known (121, 121+59, s)-Nets in Base 3

Best Known (121, 121+59, s)-Nets in Base 3

(121, 121+59, 156)-Net over F₃ — Constructive and digital

Digital (121, 180, 156)-net over F₃, using

18 times m-reduction [i] based on digital (121, 198, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 99, 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]

(121, 121+59, 307)-Net over F₃ — Digital

Digital (121, 180, 307)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(121, 121+59, 5112)-Net in Base 3 — Upper bound on s

There is no (121, 180, 5113)-net in base 3, because

1 times m-reduction [i] would yield (121, 179, 5113)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 25 396576 146812 403435 860901 712564 726323 518036 088241 156531 318647 535721 079689 363552 689539 > 3¹⁷⁹ [i]