MinT - Best Known (168−95, 168, s)-Nets in Base 8

Best Known (168−95, 168, s)-Nets in Base 8

(168−95, 168, 113)-Net over F₈ — Constructive and digital

Digital (73, 168, 113)-net over F₈, using

1 times m-reduction [i] based on digital (73, 169, 113)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 59, 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, 110, 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]

(168−95, 168, 165)-Net over F₈ — Digital

Digital (73, 168, 165)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−95, 168, 168)-Net in Base 8

(73, 168, 168)-net in base 8, using

base change [i] based on digital (31, 126, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(168−95, 168, 4215)-Net in Base 8 — Upper bound on s

There is no (73, 168, 4216)-net in base 8, because

1 times m-reduction [i] would yield (73, 167, 4216)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6 611852 543481 123043 650168 908490 733411 854873 970649 822561 844487 537256 250845 050989 202197 852164 371669 599238 666753 010299 733797 972739 732609 566077 692213 130518 > 8¹⁶⁷ [i]