MinT - Best Known (56, 56+125, s)-Nets in Base 4

Best Known (56, 56+125, s)-Nets in Base 4

(56, 56+125, 66)-Net over F₄ — Constructive and digital

Digital (56, 181, 66)-net over F₄, using

t-expansion [i] based on digital (49, 181, 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]

(56, 56+125, 91)-Net over F₄ — Digital

Digital (56, 181, 91)-net over F₄, using

t-expansion [i] based on digital (50, 181, 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]

(56, 56+125, 397)-Net in Base 4 — Upper bound on s

There is no (56, 181, 398)-net in base 4, because

1 times m-reduction [i] would yield (56, 180, 398)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 2 546827 670175 018949 421718 075519 961180 002410 226540 516678 461890 121730 128425 489306 461959 237068 535011 835900 775808 > 4¹⁸⁰ [i]