MinT - Best Known (227−149, 227, s)-Nets in Base 4

Best Known (227−149, 227, s)-Nets in Base 4

(227−149, 227, 104)-Net over F₄ — Constructive and digital

Digital (78, 227, 104)-net over F₄, using

t-expansion [i] based on digital (73, 227, 104)-net over F₄, using
- net from sequence [i] based on digital (73, 103)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 104, using
    - F₆ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(227−149, 227, 112)-Net over F₄ — Digital

Digital (78, 227, 112)-net over F₄, using

t-expansion [i] based on digital (73, 227, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(227−149, 227, 593)-Net in Base 4 — Upper bound on s

There is no (78, 227, 594)-net in base 4, because

1 times m-reduction [i] would yield (78, 226, 594)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 12446 907198 473391 718767 250785 969814 930534 419051 926662 393437 962425 042887 830825 557580 706201 781113 936762 898589 372119 796327 207804 095155 588040 > 4²²⁶ [i]