MinT - Best Known (71, 168, s)-Nets in Base 8

Best Known (71, 168, s)-Nets in Base 8

(71, 168, 110)-Net over F₈ — Constructive and digital

Digital (71, 168, 110)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 57, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]
2. digital (14, 111, 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]

(71, 168, 151)-Net over F₈ — Digital

Digital (71, 168, 151)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(71, 168, 161)-Net in Base 8

(71, 168, 161)-net in base 8, using

base change [i] based on digital (29, 126, 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]

(71, 168, 3682)-Net in Base 8 — Upper bound on s

There is no (71, 168, 3683)-net in base 8, because

1 times m-reduction [i] would yield (71, 167, 3683)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 594436 613017 117279 352147 168017 049983 839212 671816 100985 792992 129434 529721 558063 869508 057234 143933 524908 451479 293798 291358 281352 964376 449608 155571 427164 > 8¹⁶⁷ [i]