MinT - Best Known (57, 57+71, s)-Nets in Base 3

Best Known (57, 57+71, s)-Nets in Base 3

(57, 57+71, 48)-Net over F₃ — Constructive and digital

Digital (57, 128, 48)-net over F₃, using

t-expansion [i] based on digital (45, 128, 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+71, 64)-Net over F₃ — Digital

Digital (57, 128, 64)-net over F₃, using

t-expansion [i] based on digital (49, 128, 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+71, 341)-Net in Base 3 — Upper bound on s

There is no (57, 128, 342)-net in base 3, because

1 times m-reduction [i] would yield (57, 127, 342)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 4 193651 025845 876542 330724 260643 993531 751381 161040 575490 304353 > 3¹²⁷ [i]