MinT - Best Known (47, 47+89, s)-Nets in Base 4

Best Known (47, 47+89, s)-Nets in Base 4

(47, 47+89, 56)-Net over F₄ — Constructive and digital

Digital (47, 136, 56)-net over F₄, using

t-expansion [i] based on digital (33, 136, 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]

(47, 47+89, 81)-Net over F₄ — Digital

Digital (47, 136, 81)-net over F₄, using

t-expansion [i] based on digital (46, 136, 81)-net over F₄, using
- net from sequence [i] based on digital (46, 80)-sequence over F₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 46 and N(F) ≥ 81, using
    - Drinfeld modules of rank 1 [i]

(47, 47+89, 369)-Net in Base 4 — Upper bound on s

There is no (47, 136, 370)-net in base 4, because

1 times m-reduction [i] would yield (47, 135, 370)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 1977 104000 594384 003123 345986 389093 331999 793892 388438 993475 657782 726810 981744 267852 > 4¹³⁵ [i]