MinT - Best Known (115−65, 115, s)-Nets in Base 4

Best Known (115−65, 115, s)-Nets in Base 4

(115−65, 115, 66)-Net over F₄ — Constructive and digital

Digital (50, 115, 66)-net over F₄, using

t-expansion [i] based on digital (49, 115, 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]

(115−65, 115, 91)-Net over F₄ — Digital

Digital (50, 115, 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]

(115−65, 115, 569)-Net in Base 4 — Upper bound on s

There is no (50, 115, 570)-net in base 4, because

1 times m-reduction [i] would yield (50, 114, 570)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 446 075778 603418 499980 112981 446109 625541 560937 529129 639726 657064 630955 > 4¹¹⁴ [i]