MinT - Best Known (189−107, 189, s)-Nets in Base 4

Best Known (189−107, 189, s)-Nets in Base 4

(189−107, 189, 104)-Net over F₄ — Constructive and digital

Digital (82, 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]

(189−107, 189, 129)-Net over F₄ — Digital

Digital (82, 189, 129)-net over F₄, using

t-expansion [i] based on digital (81, 189, 129)-net over F₄, using
- net from sequence [i] based on digital (81, 128)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 81 and N(F) ≥ 129, using
    - Drinfeld modules of rank 1 [i]

(189−107, 189, 895)-Net in Base 4 — Upper bound on s

There is no (82, 189, 896)-net in base 4, because

1 times m-reduction [i] would yield (82, 188, 896)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 160523 346639 269596 411534 193083 342597 561693 990345 338349 286046 340561 862030 611455 931596 860133 682436 797560 280212 014389 > 4¹⁸⁸ [i]