MinT - Best Known (45, 90, s)-Nets in Base 4

Best Known (45, 90, s)-Nets in Base 4

(45, 90, 57)-Net over F₄ — Constructive and digital

Digital (45, 90, 57)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (10, 32, 27)-net over F₄, using
  - net from sequence [i] based on digital (10, 26)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 10 and N(F) ≥ 27, using
      - an explicitly constructive algebraic function field [i]
2. digital (13, 58, 30)-net over F₄, using
  - net from sequence [i] based on digital (13, 29)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 13 and N(F) ≥ 30, using
      - F₄ from the tower of function fields by GarcÃa and Stichtenoth over F₄ [i]

(45, 90, 65)-Net in Base 4 — Constructive

(45, 90, 65)-net in base 4, using

3 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, 90, 80)-Net over F₄ — Digital

Digital (45, 90, 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, 90, 805)-Net in Base 4 — Upper bound on s

There is no (45, 90, 806)-net in base 4, because

1 times m-reduction [i] would yield (45, 89, 806)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 391432 589852 254132 114899 789751 006964 291545 055307 883744 > 4⁸⁹ [i]