MinT - Best Known (89, 89+101, s)-Nets in Base 4

Best Known (89, 89+101, s)-Nets in Base 4

(89, 89+101, 104)-Net over F₄ — Constructive and digital

Digital (89, 190, 104)-net over F₄, using

t-expansion [i] based on digital (73, 190, 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, 89+101, 129)-Net over F₄ — Digital

Digital (89, 190, 129)-net over F₄, using

t-expansion [i] based on digital (81, 190, 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, 89+101, 1184)-Net in Base 4 — Upper bound on s

There is no (89, 190, 1185)-net in base 4, because

1 times m-reduction [i] would yield (89, 189, 1185)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 618380 537848 859988 202699 640587 403582 967515 468404 156477 959755 559269 360017 618232 981316 578586 028943 118911 419857 961408 > 4¹⁸⁹ [i]