MinT - Best Known (168−62, 168, s)-Nets in Base 8

Best Known (168−62, 168, s)-Nets in Base 8

(168−62, 168, 354)-Net over F₈ — Constructive and digital

Digital (106, 168, 354)-net over F₈, using

t-expansion [i] based on digital (93, 168, 354)-net over F₈, using
- 4 times m-reduction [i] based on digital (93, 172, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 86, 177)-net over F₆₄, using
    - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]

(168−62, 168, 576)-Net in Base 8 — Constructive

(106, 168, 576)-net in base 8, using

t-expansion [i] based on (105, 168, 576)-net in base 8, using
- trace code for nets [i] based on (21, 84, 288)-net in base 64, using
  - base change [i] based on digital (9, 72, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(168−62, 168, 1048)-Net over F₈ — Digital

Digital (106, 168, 1048)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(168−62, 168, 139007)-Net in Base 8 — Upper bound on s

There is no (106, 168, 139008)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 52 376031 252565 020606 713073 560208 026805 868141 560984 161066 219189 406383 760526 368960 707536 059592 286984 890951 247401 715206 231619 199884 125800 159468 894997 113745 > 8¹⁶⁸ [i]