MinT - Best Known (167−57, 167, s)-Nets in Base 3

Best Known (167−57, 167, s)-Nets in Base 3

(167−57, 167, 156)-Net over F₃ — Constructive and digital

Digital (110, 167, 156)-net over F₃, using

9 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]

(167−57, 167, 254)-Net over F₃ — Digital

Digital (110, 167, 254)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(167−57, 167, 3779)-Net in Base 3 — Upper bound on s

There is no (110, 167, 3780)-net in base 3, because

1 times m-reduction [i] would yield (110, 166, 3780)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 15 935015 522557 659929 280995 112477 603776 032870 316063 799601 475269 768569 511677 745025 > 3¹⁶⁶ [i]