MinT - Best Known (53, 53+81, s)-Nets in Base 3

Best Known (53, 53+81, s)-Nets in Base 3

(53, 53+81, 48)-Net over F₃ — Constructive and digital

Digital (53, 134, 48)-net over F₃, using

t-expansion [i] based on digital (45, 134, 48)-net over F₃, using
- net from sequence [i] based on digital (45, 47)-sequence over F₃, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃ with g(F) = 45 and N(F) ≥ 48, using
    - an explicitly constructive algebraic function field [i]

(53, 53+81, 64)-Net over F₃ — Digital

Digital (53, 134, 64)-net over F₃, using

t-expansion [i] based on digital (49, 134, 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+81, 266)-Net in Base 3 — Upper bound on s

There is no (53, 134, 267)-net in base 3, because

1 times m-reduction [i] would yield (53, 133, 267)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 2972 822163 802061 002794 720333 888475 853976 986402 128849 354904 662065 > 3¹³³ [i]