MinT - Best Known (135−65, 135, s)-Nets in Base 5

Best Known (135−65, 135, s)-Nets in Base 5

(135−65, 135, 94)-Net over F₅ — Constructive and digital

Digital (70, 135, 94)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (7, 39, 22)-net over F₅, using
  - net from sequence [i] based on digital (7, 21)-sequence over F₅, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₅ with g(F) = 7 and N(F) ≥ 22, using
      - an explicitly constructive algebraic function field [i]
2. digital (31, 96, 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]

(135−65, 135, 152)-Net over F₅ — Digital

Digital (70, 135, 152)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−65, 135, 2679)-Net in Base 5 — Upper bound on s

There is no (70, 135, 2680)-net in base 5, because

1 times m-reduction [i] would yield (70, 134, 2680)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 4645 342892 673969 648590 465303 337629 019962 523866 465188 378491 445303 741660 450737 648302 665654 595585 > 5¹³⁴ [i]