MinT - Best Known (57, 57+135, s)-Nets in Base 4

Best Known (57, 57+135, s)-Nets in Base 4

(57, 57+135, 66)-Net over F₄ — Constructive and digital

Digital (57, 192, 66)-net over F₄, using

t-expansion [i] based on digital (49, 192, 66)-net over F₄, using
- net from sequence [i] based on digital (49, 65)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 49 and N(F) ≥ 66, using
    - T₆ from the second tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(57, 57+135, 91)-Net over F₄ — Digital

Digital (57, 192, 91)-net over F₄, using

t-expansion [i] based on digital (50, 192, 91)-net over F₄, using
- net from sequence [i] based on digital (50, 90)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 50 and N(F) ≥ 91, using
    - a curve from the manYPoints database [i]

(57, 57+135, 394)-Net in Base 4 — Upper bound on s

There is no (57, 192, 395)-net in base 4, because

1 times m-reduction [i] would yield (57, 191, 395)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10 789818 014533 003173 681785 924554 906993 684247 598506 472706 430256 530513 941695 134022 958445 487981 794286 429607 147723 674952 > 4¹⁹¹ [i]