MinT - Best Known (245−141, 245, s)-Nets in Base 4

Best Known (245−141, 245, s)-Nets in Base 4

(245−141, 245, 104)-Net over F₄ — Constructive and digital

Digital (104, 245, 104)-net over F₄, using

t-expansion [i] based on digital (73, 245, 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]

(245−141, 245, 144)-Net over F₄ — Digital

Digital (104, 245, 144)-net over F₄, using

t-expansion [i] based on digital (91, 245, 144)-net over F₄, using
- net from sequence [i] based on digital (91, 143)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 91 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(245−141, 245, 1068)-Net in Base 4 — Upper bound on s

There is no (104, 245, 1069)-net in base 4, because

1 times m-reduction [i] would yield (104, 244, 1069)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 832 633094 448982 134192 098511 701127 246122 815979 701722 866555 972714 580675 716642 594872 729317 600690 189156 441765 895552 868790 440499 897520 697947 106069 432512 > 4²⁴⁴ [i]