MinT - Best Known (68, 68+40, s)-Nets in Base 25

Best Known (68, 68+40, s)-Nets in Base 25

(68, 68+40, 378)-Net over F₂₅ — Constructive and digital

Digital (68, 108, 378)-net over F₂₅, using

(u,Â u+v)-construction [i] based on
1. digital (23, 43, 178)-net over F₂₅, using
  - (u,Â u+v)-construction [i] based on
    1. digital (3, 13, 52)-net over F₂₅, using
      - net from sequence [i] based on digital (3, 51)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 3 and N(F) ≥ 52, using
        a function field by GarcÃa and Quoos [i]
    2. digital (10, 30, 126)-net over F₂₅, using
      - net from sequence [i] based on digital (10, 125)-sequence over F₂₅, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 10 and N(F) ≥ 126, using
        the Hermitian function field over F₂₅ [i]
2. digital (25, 65, 200)-net over F₂₅, using
  - net from sequence [i] based on digital (25, 199)-sequence over F₂₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅ with g(F) = 25 and N(F) ≥ 200, using
      - a function field by GarcÃa and Quoos [i]

(68, 68+40, 4789)-Net over F₂₅ — Digital

Digital (68, 108, 4789)-net over F₂₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(68, 68+40, large)-Net in Base 25 — Upper bound on s

There is no (68, 108, large)-net in base 25, because

38 times m-reduction [i] would yield (68, 70, large)-net in base 25, but
- mutually orthogonal hypercube bound [i]