MinT - Best Known (57, 57+115, s)-Nets in Base 4

Best Known (57, 57+115, s)-Nets in Base 4

(57, 57+115, 66)-Net over F₄ — Constructive and digital

Digital (57, 172, 66)-net over F₄, using

t-expansion [i] based on digital (49, 172, 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+115, 91)-Net over F₄ — Digital

Digital (57, 172, 91)-net over F₄, using

t-expansion [i] based on digital (50, 172, 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+115, 425)-Net in Base 4 — Upper bound on s

There is no (57, 172, 426)-net in base 4, because

1 times m-reduction [i] would yield (57, 171, 426)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 9 246143 990260 127666 921384 619416 411515 613205 821656 299330 173641 452432 869568 320075 449109 083271 215204 126072 > 4¹⁷¹ [i]