MinT - Best Known (52, 52+47, s)-Nets in Base 3

Best Known (52, 52+47, s)-Nets in Base 3

(52, 52+47, 52)-Net over F₃ — Constructive and digital

Digital (52, 99, 52)-net over F₃, using

1 times m-reduction [i] based on digital (52, 100, 52)-net over F₃, using
- (u,Â u+v)-construction [i] based on
  1. digital (13, 37, 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, 63, 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]

(52, 52+47, 65)-Net over F₃ — Digital

Digital (52, 99, 65)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+47, 486)-Net in Base 3 — Upper bound on s

There is no (52, 99, 487)-net in base 3, because

1 times m-reduction [i] would yield (52, 98, 487)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 58552 725068 438012 657153 118030 472681 758626 474739 > 3⁹⁸ [i]