MinT - Best Known (70, 70+67, s)-Nets in Base 5

Best Known (70, 70+67, s)-Nets in Base 5

(70, 70+67, 93)-Net over F₅ — Constructive and digital

Digital (70, 137, 93)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (6, 39, 21)-net over F₅, using
  - net from sequence [i] based on digital (6, 20)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 6 and N(F) ≥ 21, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 98, 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+67, 145)-Net over F₅ — Digital

Digital (70, 137, 145)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(70, 70+67, 2475)-Net in Base 5 — Upper bound on s

There is no (70, 137, 2476)-net in base 5, because

1 times m-reduction [i] would yield (70, 136, 2476)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 115416 393611 548061 475962 373017 625834 039671 770537 969746 956208 221833 253696 387216 170400 091322 434225 > 5¹³⁶ [i]