MinT - Best Known (67, 67+85, s)-Nets in Base 8

Best Known (67, 67+85, s)-Nets in Base 8

(67, 67+85, 113)-Net over F₈ — Constructive and digital

Digital (67, 152, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 53, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (14, 99, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(67, 67+85, 158)-Net over F₈ — Digital

Digital (67, 152, 158)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+85, 161)-Net in Base 8

(67, 152, 161)-net in base 8, using

base change [i] based on digital (29, 114, 161)-net over F₁₆, using
- net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
    - a curve from the manYPoints database [i]

(67, 67+85, 4137)-Net in Base 8 — Upper bound on s

There is no (67, 152, 4138)-net in base 8, because

1 times m-reduction [i] would yield (67, 151, 4138)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 23326 810034 177162 577423 545213 220438 951279 057257 831892 110650 333991 971713 502405 908128 540933 202399 933972 253158 817354 800133 362493 646804 104816 > 8¹⁵¹ [i]