MinT - Best Known (57−35, 57, s)-Nets in Base 4

Best Known (57−35, 57, s)-Nets in Base 4

(57−35, 57, 34)-Net over F₄ — Constructive and digital

Digital (22, 57, 34)-net over F₄, using

t-expansion [i] based on digital (21, 57, 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]

(57−35, 57, 44)-Net over F₄ — Digital

Digital (22, 57, 44)-net over F₄, using

t-expansion [i] based on digital (21, 57, 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]

(57−35, 57, 216)-Net in Base 4 — Upper bound on s

There is no (22, 57, 217)-net in base 4, because

1 times m-reduction [i] would yield (22, 56, 217)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5259 585819 801971 517431 779469 441140 > 4⁵⁶ [i]