MinT - Best Known (68, 146, s)-Nets in Base 3

Best Known (68, 146, s)-Nets in Base 3

(68, 146, 52)-Net over F₃ — Constructive and digital

Digital (68, 146, 52)-net over F₃, using

2 times m-reduction [i] based on digital (68, 148, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 53, 24)-net over F₃, using
    - net from sequence [i] based on digital (13, 23)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 13 and N(F) ≥ 24, using
        a function field by SÃ©mirat [i]
  2. digital (15, 95, 28)-net over F₃, using
    - net from sequence [i] based on digital (15, 27)-sequence over F₃, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 15 and N(F) ≥ 28, using
        an explicitly constructive algebraic function field [i]

(68, 146, 72)-Net over F₃ — Digital

Digital (68, 146, 72)-net over F₃, using

t-expansion [i] based on digital (67, 146, 72)-net over F₃, using
- net from sequence [i] based on digital (67, 71)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 67 and N(F) ≥ 72, using
    - algebraic function fields over â„¤₃ by Niederreiter/Xing [i]

(68, 146, 433)-Net in Base 3 — Upper bound on s

There is no (68, 146, 434)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4865 792437 948307 646512 401131 451125 844530 301985 354065 306142 960121 294441 > 3¹⁴⁶ [i]