MinT - Best Known (45, 45+71, s)-Nets in Base 4

Best Known (45, 45+71, s)-Nets in Base 4

(45, 45+71, 56)-Net over F₄ — Constructive and digital

Digital (45, 116, 56)-net over F₄, using

t-expansion [i] based on digital (33, 116, 56)-net over F₄, using
- net from sequence [i] based on digital (33, 55)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 33 and N(F) ≥ 56, using
    - F₅ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(45, 45+71, 80)-Net over F₄ — Digital

Digital (45, 116, 80)-net over F₄, using

net from sequence [i] based on digital (45, 79)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 45 and N(F) ≥ 80, using
  - Drinfeld modules of rank 1 [i]

(45, 45+71, 412)-Net in Base 4 — Upper bound on s

There is no (45, 116, 413)-net in base 4, because

1 times m-reduction [i] would yield (45, 115, 413)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1735 778978 455868 531503 227285 167453 330868 441206 807601 931760 917156 766416 > 4¹¹⁵ [i]