MinT - Best Known (58−37, 58, s)-Nets in Base 4

Best Known (58−37, 58, s)-Nets in Base 4

(58−37, 58, 34)-Net over F₄ — Constructive and digital

Digital (21, 58, 34)-net over F₄, using

net from sequence [i] based on digital (21, 33)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 34, using
  - T₅ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(58−37, 58, 44)-Net over F₄ — Digital

Digital (21, 58, 44)-net over F₄, using

net from sequence [i] based on digital (21, 43)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
  - a curve from the manYPoints database [i]

(58−37, 58, 188)-Net in Base 4 — Upper bound on s

There is no (21, 58, 189)-net in base 4, because

1 times m-reduction [i] would yield (21, 57, 189)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 20946 429061 896852 133758 463234 075210 > 4⁵⁷ [i]