MinT - Best Known (45, 45+47, s)-Nets in Base 4

Best Known (45, 45+47, s)-Nets in Base 4

(45, 45+47, 56)-Net over F₄ — Constructive and digital

Digital (45, 92, 56)-net over F₄, using

t-expansion [i] based on digital (33, 92, 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+47, 65)-Net in Base 4 — Constructive

(45, 92, 65)-net in base 4, using

1 times m-reduction [i] based on (45, 93, 65)-net in base 4, using
- base change [i] based on digital (14, 62, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(45, 45+47, 80)-Net over F₄ — Digital

Digital (45, 92, 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+47, 739)-Net in Base 4 — Upper bound on s

There is no (45, 92, 740)-net in base 4, because

1 times m-reduction [i] would yield (45, 91, 740)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 6 311939 806436 637680 296092 609334 452828 242803 032057 807736 > 4⁹¹ [i]