MinT - Best Known (70, 70+69, s)-Nets in Base 5

Best Known (70, 70+69, s)-Nets in Base 5

(70, 70+69, 92)-Net over F₅ — Constructive and digital

Digital (70, 139, 92)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 39, 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, 100, 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]

(70, 70+69, 140)-Net over F₅ — Digital

Digital (70, 139, 140)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+69, 2300)-Net in Base 5 — Upper bound on s

There is no (70, 139, 2301)-net in base 5, because

1 times m-reduction [i] would yield (70, 138, 2301)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 2 910708 129062 319260 463052 443143 217365 333693 535521 367867 377962 418755 084986 248865 098008 030176 636825 > 5¹³⁸ [i]