MinT - Best Known (189−111, 189, s)-Nets in Base 4

Best Known (189−111, 189, s)-Nets in Base 4

(189−111, 189, 104)-Net over F₄ — Constructive and digital

Digital (78, 189, 104)-net over F₄, using

t-expansion [i] based on digital (73, 189, 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]

(189−111, 189, 112)-Net over F₄ — Digital

Digital (78, 189, 112)-net over F₄, using

t-expansion [i] based on digital (73, 189, 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]

(189−111, 189, 768)-Net in Base 4 — Upper bound on s

There is no (78, 189, 769)-net in base 4, because

1 times m-reduction [i] would yield (78, 188, 769)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 160165 076208 720193 978825 176920 108068 623410 148181 494792 917778 321662 811596 695151 735602 636919 343873 257807 860361 060608 > 4¹⁸⁸ [i]