MinT - Best Known (140−89, 140, s)-Nets in Base 4

Best Known (140−89, 140, s)-Nets in Base 4

(140−89, 140, 66)-Net over F₄ — Constructive and digital

Digital (51, 140, 66)-net over F₄, using

t-expansion [i] based on digital (49, 140, 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]

(140−89, 140, 91)-Net over F₄ — Digital

Digital (51, 140, 91)-net over F₄, using

t-expansion [i] based on digital (50, 140, 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]

(140−89, 140, 423)-Net in Base 4 — Upper bound on s

There is no (51, 140, 424)-net in base 4, because

1 times m-reduction [i] would yield (51, 139, 424)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 493664 069535 616972 936444 853932 636719 285252 730210 198505 609360 107582 336241 082252 366928 > 4¹³⁹ [i]