MinT - Best Known (136−67, 136, s)-Nets in Base 5

Best Known (136−67, 136, s)-Nets in Base 5

(136−67, 136, 92)-Net over F₅ — Constructive and digital

Digital (69, 136, 92)-net over F₅, using

(u,Â u+v)-construction [i] based on
1. digital (5, 38, 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, 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]

(136−67, 136, 141)-Net over F₅ — Digital

Digital (69, 136, 141)-net over F₅, using

Gilbertâ€“VarÅ¡amov bound [i]

(136−67, 136, 2356)-Net in Base 5 — Upper bound on s

There is no (69, 136, 2357)-net in base 5, because

1 times m-reduction [i] would yield (69, 135, 2357)-net in base 5, but
- the generalized Rao bound for nets shows that 5^mÂ â‰¥ 23084 998078 569920 656559 650807 390002 575912 061860 123461 413027 036508 536899 110752 254709 776855 319125 > 5¹³⁵ [i]