MinT - Best Known (62−31, 62, s)-Nets in Base 4

Best Known (62−31, 62, s)-Nets in Base 4

(62−31, 62, 44)-Net over F₄ — Constructive and digital

Digital (31, 62, 44)-net over F₄, using

1 times m-reduction [i] based on digital (31, 63, 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, 42, 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]

(62−31, 62, 46)-Net in Base 4 — Constructive

(31, 62, 46)-net in base 4, using

1 times m-reduction [i] based on (31, 63, 46)-net in base 4, using
- base change [i] based on digital (10, 42, 46)-net over F₈, using
  - net from sequence [i] based on digital (10, 45)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 9, N(F)Â =Â 45, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using an explicitly constructive algebraic function field [i]

(62−31, 62, 60)-Net over F₄ — Digital

Digital (31, 62, 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]

(62−31, 62, 589)-Net in Base 4 — Upper bound on s

There is no (31, 62, 590)-net in base 4, because

1 times m-reduction [i] would yield (31, 61, 590)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5 410901 279692 620888 945328 141152 916240 > 4⁶¹ [i]