MinT - Best Known (70, 70+71, s)-Nets in Base 5

Best Known (70, 70+71, s)-Nets in Base 5

(70, 70+71, 90)-Net over F₅ — Constructive and digital

Digital (70, 141, 90)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (4, 39, 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, 102, 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+71, 135)-Net over F₅ — Digital

Digital (70, 141, 135)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+71, 2147)-Net in Base 5 — Upper bound on s

There is no (70, 141, 2148)-net in base 5, because

1 times m-reduction [i] would yield (70, 140, 2148)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 72 351095 551017 831231 850114 659530 555706 719826 669705 286351 554774 253875 863978 272252 396049 015965 368465 > 5¹⁴⁰ [i]