MinT - Best Known (87, 87+107, s)-Nets in Base 4

Best Known (87, 87+107, s)-Nets in Base 4

(87, 87+107, 104)-Net over F₄ — Constructive and digital

Digital (87, 194, 104)-net over F₄, using

t-expansion [i] based on digital (73, 194, 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+107, 129)-Net over F₄ — Digital

Digital (87, 194, 129)-net over F₄, using

t-expansion [i] based on digital (81, 194, 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+107, 1026)-Net in Base 4 — Upper bound on s

There is no (87, 194, 1027)-net in base 4, because

1 times m-reduction [i] would yield (87, 193, 1027)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 163 618474 284016 989271 123040 328248 248693 290197 588325 075091 884481 604472 417305 748817 501555 128503 122045 657041 823921 968640 > 4¹⁹³ [i]