MinT - Best Known (42−21, 42, s)-Nets in Base 4

Best Known (42−21, 42, s)-Nets in Base 4

(42−21, 42, 36)-Net over F₄ — Constructive and digital

Digital (21, 42, 36)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (4, 14, 15)-net over F₄, using
  - net from sequence [i] based on digital (4, 14)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 4 and N(F) ≥ 15, using
      - an explicitly constructive algebraic function field [i]
2. digital (7, 28, 21)-net over F₄, using
  - net from sequence [i] based on digital (7, 20)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 7 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]

(42−21, 42, 44)-Net over F₄ — Digital

Digital (21, 42, 44)-net over F₄, using

net from sequence [i] based on digital (21, 43)-sequence over F₄, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 21 and N(F) ≥ 44, using
  - a curve from the manYPoints database [i]

(42−21, 42, 436)-Net in Base 4 — Upper bound on s

There is no (21, 42, 437)-net in base 4, because

1 times m-reduction [i] would yield (21, 41, 437)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 4 930574 081043 855696 931708 > 4⁴¹ [i]