MinT - Best Known (58, 58+129, s)-Nets in Base 4

Best Known (58, 58+129, s)-Nets in Base 4

(58, 58+129, 66)-Net over F₄ — Constructive and digital

Digital (58, 187, 66)-net over F₄, using

t-expansion [i] based on digital (49, 187, 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]

(58, 58+129, 91)-Net over F₄ — Digital

Digital (58, 187, 91)-net over F₄, using

t-expansion [i] based on digital (50, 187, 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]

(58, 58+129, 411)-Net in Base 4 — Upper bound on s

There is no (58, 187, 412)-net in base 4, because

1 times m-reduction [i] would yield (58, 186, 412)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10209 100063 073785 360190 731118 842746 330118 027107 214106 940131 895367 926210 747654 919924 652785 439175 484000 018661 719690 > 4¹⁸⁶ [i]