MinT - Best Known (137−72, 137, s)-Nets in Base 3

Best Known (137−72, 137, s)-Nets in Base 3

(137−72, 137, 52)-Net over F₃ — Constructive and digital

Digital (65, 137, 52)-net over F₃, using

2 times m-reduction [i] based on digital (65, 139, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 50, 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, 89, 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]

(137−72, 137, 64)-Net over F₃ — Digital

Digital (65, 137, 64)-net over F₃, using

t-expansion [i] based on digital (49, 137, 64)-net over F₃, using
- net from sequence [i] based on digital (49, 63)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 49 and N(F) ≥ 64, using
    - a curve from the manYPoints database [i]

(137−72, 137, 432)-Net in Base 3 — Upper bound on s

There is no (65, 137, 433)-net in base 3, because

the generalized Rao bound for nets shows that 3^mÂ â‰¥ 237676 261947 554270 132412 686835 058903 619633 921431 020410 111444 372049 > 3¹³⁷ [i]