MinT - Best Known (72, 227, s)-Nets in Base 4

Best Known (72, 227, s)-Nets in Base 4

(72, 227, 66)-Net over F₄ — Constructive and digital

Digital (72, 227, 66)-net over F₄, using

t-expansion [i] based on digital (49, 227, 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]

(72, 227, 105)-Net over F₄ — Digital

Digital (72, 227, 105)-net over F₄, using

t-expansion [i] based on digital (70, 227, 105)-net over F₄, using
- net from sequence [i] based on digital (70, 104)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 70 and N(F) ≥ 105, using
    - Drinfeld modules of rank 1 [i]

(72, 227, 513)-Net in Base 4 — Upper bound on s

There is no (72, 227, 514)-net in base 4, because

1 times m-reduction [i] would yield (72, 226, 514)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 11969 273606 165454 302127 880021 769071 271035 926473 659942 458986 562839 197603 135310 011918 790180 818823 878487 655571 300205 650098 672739 651310 158720 > 4²²⁶ [i]