MinT - Best Known (56, 56+123, s)-Nets in Base 4

Best Known (56, 56+123, s)-Nets in Base 4

(56, 56+123, 66)-Net over F₄ — Constructive and digital

Digital (56, 179, 66)-net over F₄, using

t-expansion [i] based on digital (49, 179, 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+123, 91)-Net over F₄ — Digital

Digital (56, 179, 91)-net over F₄, using

t-expansion [i] based on digital (50, 179, 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+123, 400)-Net in Base 4 — Upper bound on s

There is no (56, 179, 401)-net in base 4, because

1 times m-reduction [i] would yield (56, 178, 401)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 159759 181781 546109 991063 586460 958189 281581 174230 159468 142547 226041 590847 685527 156044 295058 000183 311856 737280 > 4¹⁷⁸ [i]