MinT - Best Known (56−41, 56, s)-Nets in Base 4

Best Known (56−41, 56, s)-Nets in Base 4

(56−41, 56, 33)-Net over F₄ — Constructive and digital

Digital (15, 56, 33)-net over F₄, using

net from sequence [i] based on digital (15, 32)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 33, using
  - an explicitly constructive algebraic function field [i]

(56−41, 56, 35)-Net over F₄ — Digital

Digital (15, 56, 35)-net over F₄, using

net from sequence [i] based on digital (15, 34)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 15 and N(F) ≥ 35, using
  - a curve from the manYPoints database [i]

(56−41, 56, 76)-Net in Base 4 — Upper bound on s

There is no (15, 56, 77)-net in base 4, because

extracting embedded orthogonal array [i] would yield OA(4⁵⁶, 77, S₄, 41), but
- the linear programming bound shows that MÂ â‰¥ 2852 409624 643877 869804 773184 242737 249513 373696 / 530979 549763 > 4⁵⁶ [i]