MinT - Best Known (43−21, 43, s)-Nets in Base 4

Best Known (43−21, 43, s)-Nets in Base 4

(43−21, 43, 38)-Net over F₄ — Constructive and digital

Digital (22, 43, 38)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (5, 15, 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 (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]

(43−21, 43, 48)-Net over F₄ — Digital

Digital (22, 43, 48)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(43−21, 43, 502)-Net in Base 4 — Upper bound on s

There is no (22, 43, 503)-net in base 4, because

1 times m-reduction [i] would yield (22, 42, 503)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 19 671754 582371 420292 544746 > 4⁴² [i]