MinT - Best Known (18, 18+17, s)-Nets in Base 4

Best Known (18, 18+17, s)-Nets in Base 4

(18, 18+17, 35)-Net over F₄ — Constructive and digital

Digital (18, 35, 35)-net over F₄, using

(u,Â u+v)-construction [i] based on
1. digital (3, 11, 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 (7, 24, 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]

(18, 18+17, 43)-Net over F₄ — Digital

Digital (18, 35, 43)-net over F₄, using

Gilbertâ€“VarÅ¡amov bound [i]

(18, 18+17, 448)-Net in Base 4 — Upper bound on s

There is no (18, 35, 449)-net in base 4, because

1 times m-reduction [i] would yield (18, 34, 449)-net in base 4, but
- the generalized Rao bound for nets shows that 4^mÂ â‰¥ 299 613848 582692 952392 > 4³⁴ [i]