MinT - Best Known (112−57, 112, s)-Nets in Base 3

Best Known (112−57, 112, s)-Nets in Base 3

(112−57, 112, 48)-Net over F₃ — Constructive and digital

Digital (55, 112, 48)-net over F₃, using

t-expansion [i] based on digital (45, 112, 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]

(112−57, 112, 64)-Net over F₃ — Digital

Digital (55, 112, 64)-net over F₃, using

t-expansion [i] based on digital (49, 112, 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]

(112−57, 112, 413)-Net in Base 3 — Upper bound on s

There is no (55, 112, 414)-net in base 3, because

1 times m-reduction [i] would yield (55, 111, 414)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 96304 987964 441510 758750 114915 106596 877158 458723 356025 > 3¹¹¹ [i]