MinT - Best Known (66, 66+63, s)-Nets in Base 5

Best Known (66, 66+63, s)-Nets in Base 5

(66, 66+63, 90)-Net over F₅ — Constructive and digital

Digital (66, 129, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 35, 18)-net over F₅, using
  - net from sequence [i] based on digital (4, 17)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 4 and N(F) ≥ 18, 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]

(66, 66+63, 139)-Net over F₅ — Digital

Digital (66, 129, 139)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+63, 2365)-Net in Base 5 — Upper bound on s

There is no (66, 129, 2366)-net in base 5, because

1 times m-reduction [i] would yield (66, 128, 2366)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 296274 638645 435700 450743 932919 550263 545026 531108 663605 350054 356514 861232 804217 223760 042265 > 5¹²⁸ [i]