MinT - Best Known (187−113, 187, s)-Nets in Base 4

Best Known (187−113, 187, s)-Nets in Base 4

(187−113, 187, 104)-Net over F₄ — Constructive and digital

Digital (74, 187, 104)-net over F₄, using

t-expansion [i] based on digital (73, 187, 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]

(187−113, 187, 112)-Net over F₄ — Digital

Digital (74, 187, 112)-net over F₄, using

t-expansion [i] based on digital (73, 187, 112)-net over F₄, using
- net from sequence [i] based on digital (73, 111)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 73 and N(F) ≥ 112, using
    - Drinfeld modules of rank 1 [i]

(187−113, 187, 678)-Net in Base 4 — Upper bound on s

There is no (74, 187, 679)-net in base 4, because

1 times m-reduction [i] would yield (74, 186, 679)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 10349 004287 091230 553903 389740 237049 466793 964585 446394 116491 611913 952801 147677 291494 812375 542342 393315 177801 426256 > 4¹⁸⁶ [i]