MinT - Best Known (67, 67+63, s)-Nets in Base 5

Best Known (67, 67+63, s)-Nets in Base 5

(67, 67+63, 92)-Net over F₅ — Constructive and digital

Digital (67, 130, 92)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 36, 20)-net over F₅, using
  - net from sequence [i] based on digital (5, 19)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 5 and N(F) ≥ 20, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 94, 72)-net over F₅, using
  - net from sequence [i] based on digital (31, 71)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 31 and N(F) ≥ 72, using
      - an explicitly constructive algebraic function field [i]

(67, 67+63, 144)-Net over F₅ — Digital

Digital (67, 130, 144)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+63, 2492)-Net in Base 5 — Upper bound on s

There is no (67, 130, 2493)-net in base 5, because

1 times m-reduction [i] would yield (67, 129, 2493)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 1 476173 603934 885029 646605 447343 408237 301290 585434 039903 835559 192510 979351 443623 104635 051261 > 5¹²⁹ [i]