MinT - Best Known (53, 53+49, s)-Nets in Base 3

Best Known (53, 53+49, s)-Nets in Base 3

(53, 53+49, 52)-Net over F₃ — Constructive and digital

Digital (53, 102, 52)-net over F₃, using

1 times m-reduction [i] based on digital (53, 103, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 38, 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, 65, 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]

(53, 53+49, 64)-Net over F₃ — Digital

Digital (53, 102, 64)-net over F₃, using

t-expansion [i] based on digital (49, 102, 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]

(53, 53+49, 476)-Net in Base 3 — Upper bound on s

There is no (53, 102, 477)-net in base 3, because

1 times m-reduction [i] would yield (53, 101, 477)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 1 615591 140411 436234 213884 443266 887996 549700 579617 > 3¹⁰¹ [i]