MinT - Best Known (89, 198, s)-Nets in Base 4

Best Known (89, 198, s)-Nets in Base 4

(89, 198, 104)-Net over F₄ — Constructive and digital

Digital (89, 198, 104)-net over F₄, using

t-expansion [i] based on digital (73, 198, 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]

(89, 198, 129)-Net over F₄ — Digital

Digital (89, 198, 129)-net over F₄, using

t-expansion [i] based on digital (81, 198, 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]

(89, 198, 1054)-Net in Base 4 — Upper bound on s

There is no (89, 198, 1055)-net in base 4, because

1 times m-reduction [i] would yield (89, 197, 1055)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 40867 414586 442368 162490 507990 670423 592089 555683 210160 228634 322199 775707 361156 358947 154890 530370 512866 214360 404309 145888 > 4¹⁹⁷ [i]