MinT - Best Known (136−81, 136, s)-Nets in Base 4

Best Known (136−81, 136, s)-Nets in Base 4

(136−81, 136, 66)-Net over F₄ — Constructive and digital

Digital (55, 136, 66)-net over F₄, using

t-expansion [i] based on digital (49, 136, 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]

(136−81, 136, 91)-Net over F₄ — Digital

Digital (55, 136, 91)-net over F₄, using

t-expansion [i] based on digital (50, 136, 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]

(136−81, 136, 533)-Net in Base 4 — Upper bound on s

There is no (55, 136, 534)-net in base 4, because

1 times m-reduction [i] would yield (55, 135, 534)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1948 484050 315778 103637 402022 509927 326431 443858 726038 640486 986451 169350 949884 355296 > 4¹³⁵ [i]