MinT - Best Known (219−131, 219, s)-Nets in Base 4

Best Known (219−131, 219, s)-Nets in Base 4

(219−131, 219, 104)-Net over F₄ — Constructive and digital

Digital (88, 219, 104)-net over F₄, using

t-expansion [i] based on digital (73, 219, 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]

(219−131, 219, 129)-Net over F₄ — Digital

Digital (88, 219, 129)-net over F₄, using

t-expansion [i] based on digital (81, 219, 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]

(219−131, 219, 820)-Net in Base 4 — Upper bound on s

There is no (88, 219, 821)-net in base 4, because

1 times m-reduction [i] would yield (88, 218, 821)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 189200 427043 053963 150035 385740 779065 927862 131804 800990 691854 243305 100078 907925 668000 260891 397460 293842 952812 769314 069668 640237 734528 > 4²¹⁸ [i]