MinT - Best Known (57, 57+61, s)-Nets in Base 3

Best Known (57, 57+61, s)-Nets in Base 3

(57, 57+61, 48)-Net over F₃ — Constructive and digital

Digital (57, 118, 48)-net over F₃, using

t-expansion [i] based on digital (45, 118, 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]

(57, 57+61, 64)-Net over F₃ — Digital

Digital (57, 118, 64)-net over F₃, using

t-expansion [i] based on digital (49, 118, 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]

(57, 57+61, 408)-Net in Base 3 — Upper bound on s

There is no (57, 118, 409)-net in base 3, because

1 times m-reduction [i] would yield (57, 117, 409)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 69 583075 426121 925692 146611 749515 750180 419503 238002 831145 > 3¹¹⁷ [i]