MinT - Best Known (64−33, 64, s)-Nets in Base 4

Best Known (64−33, 64, s)-Nets in Base 4

(64−33, 64, 44)-Net over F₄ — Constructive and digital

Digital (31, 64, 44)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 21, 17)-net over F₄, using
  - net from sequence [i] based on digital (5, 16)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 5 and N(F) ≥ 17, using
      - an explicitly constructive algebraic function field [i]
2. digital (10, 43, 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]

(64−33, 64, 45)-Net in Base 4 — Constructive

(31, 64, 45)-net in base 4, using

2 times m-reduction [i] based on (31, 66, 45)-net in base 4, using
- base change [i] based on digital (9, 44, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(64−33, 64, 60)-Net over F₄ — Digital

Digital (31, 64, 60)-net over F₄, using

net from sequence [i] based on digital (31, 59)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 31 and N(F) ≥ 60, using
  - Drinfeld modules of rank 1 [i]

(64−33, 64, 519)-Net in Base 4 — Upper bound on s

There is no (31, 64, 520)-net in base 4, because

1 times m-reduction [i] would yield (31, 63, 520)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 86 543693 116518 491873 769093 608554 660192 > 4⁶³ [i]