MinT - Best Known (123, 123+72, s)-Nets in Base 3

Best Known (123, 123+72, s)-Nets in Base 3

(123, 123+72, 156)-Net over F₃ — Constructive and digital

Digital (123, 195, 156)-net over F₃, using

7 times m-reduction [i] based on digital (123, 202, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 101, 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]

(123, 123+72, 230)-Net over F₃ — Digital

Digital (123, 195, 230)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(123, 123+72, 2706)-Net in Base 3 — Upper bound on s

There is no (123, 195, 2707)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1093 294466 256727 061000 619371 842906 659520 647551 305950 336364 540950 146302 676046 965036 008721 601945 > 3¹⁹⁵ [i]