MinT - Best Known (87, 87+103, s)-Nets in Base 4

Best Known (87, 87+103, s)-Nets in Base 4

(87, 87+103, 104)-Net over F₄ — Constructive and digital

Digital (87, 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]

(87, 87+103, 129)-Net over F₄ — Digital

Digital (87, 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]

(87, 87+103, 1085)-Net in Base 4 — Upper bound on s

There is no (87, 190, 1086)-net in base 4, because

1 times m-reduction [i] would yield (87, 189, 1086)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 629178 620584 283841 136056 135690 057456 448804 086528 491811 027240 455511 054170 460028 276389 090913 753341 451396 579902 349390 > 4¹⁸⁹ [i]