MinT - Best Known (190−131, 190, s)-Nets in Base 4

Best Known (190−131, 190, s)-Nets in Base 4

(190−131, 190, 66)-Net over F₄ — Constructive and digital

Digital (59, 190, 66)-net over F₄, using

t-expansion [i] based on digital (49, 190, 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]

(190−131, 190, 91)-Net over F₄ — Digital

Digital (59, 190, 91)-net over F₄, using

t-expansion [i] based on digital (50, 190, 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]

(190−131, 190, 418)-Net in Base 4 — Upper bound on s

There is no (59, 190, 419)-net in base 4, because

1 times m-reduction [i] would yield (59, 189, 419)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 646391 984108 543920 024351 620215 804779 412473 410720 805072 204722 832196 326078 403605 389156 633093 023128 184846 720933 549500 > 4¹⁸⁹ [i]