MinT - Best Known (57, 57+119, s)-Nets in Base 4

Best Known (57, 57+119, s)-Nets in Base 4

(57, 57+119, 66)-Net over F₄ — Constructive and digital

Digital (57, 176, 66)-net over F₄, using

t-expansion [i] based on digital (49, 176, 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+119, 91)-Net over F₄ — Digital

Digital (57, 176, 91)-net over F₄, using

t-expansion [i] based on digital (50, 176, 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+119, 417)-Net in Base 4 — Upper bound on s

There is no (57, 176, 418)-net in base 4, because

1 times m-reduction [i] would yield (57, 175, 418)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2387 480720 230660 068912 380889 547875 040333 210459 895098 998483 562849 381776 142290 069473 885554 632580 658550 454720 > 4¹⁷⁵ [i]