MinT - Best Known (199−113, 199, s)-Nets in Base 4

Best Known (199−113, 199, s)-Nets in Base 4

(199−113, 199, 104)-Net over F₄ — Constructive and digital

Digital (86, 199, 104)-net over F₄, using

t-expansion [i] based on digital (73, 199, 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]

(199−113, 199, 129)-Net over F₄ — Digital

Digital (86, 199, 129)-net over F₄, using

t-expansion [i] based on digital (81, 199, 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]

(199−113, 199, 927)-Net in Base 4 — Upper bound on s

There is no (86, 199, 928)-net in base 4, because

1 times m-reduction [i] would yield (86, 198, 928)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 162176 421967 321019 596837 626638 875969 473486 764978 121052 884917 466752 548744 079483 396918 362358 883151 187737 874354 076956 253460 > 4¹⁹⁸ [i]