MinT - Best Known (47−23, 47, s)-Nets in Base 4

Best Known (47−23, 47, s)-Nets in Base 4

(47−23, 47, 41)-Net over F₄ — Constructive and digital

Digital (24, 47, 41)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 14)-net over F₄, using
  - net from sequence [i] based on digital (3, 13)-sequence over F₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₄ with g(F) = 3 and N(F) ≥ 14, using
      - an explicitly constructive algebraic function field [i]
2. digital (10, 33, 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]

(47−23, 47, 50)-Net over F₄ — Digital

Digital (24, 47, 50)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(47−23, 47, 530)-Net in Base 4 — Upper bound on s

There is no (24, 47, 531)-net in base 4, because

1 times m-reduction [i] would yield (24, 46, 531)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 5012 798325 347662 867073 200600 > 4⁴⁶ [i]