MinT - Best Known (57, 57+55, s)-Nets in Base 3

Best Known (57, 57+55, s)-Nets in Base 3

(57, 57+55, 56)-Net over F₃ — Constructive and digital

Digital (57, 112, 56)-net over F₃, using

(u,Â u+v)-construction [i] based on
1. digital (15, 42, 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]
2. digital (15, 70, 28)-net over F₃, using
  - net from sequence [i] based on digital (15, 27)-sequence over F₃ (see above)

(57, 57+55, 65)-Net over F₃ — Digital

Digital (57, 112, 65)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+55, 473)-Net in Base 3 — Upper bound on s

There is no (57, 112, 474)-net in base 3, because

1 times m-reduction [i] would yield (57, 111, 474)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 91487 723391 686642 832481 078394 959573 326931 470055 270273 > 3¹¹¹ [i]