MinT - Best Known (108−32, 108, s)-Nets in Base 8

Best Known (108−32, 108, s)-Nets in Base 8

(108−32, 108, 419)-Net over F₈ — Constructive and digital

Digital (76, 108, 419)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (14, 30, 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]
2. digital (46, 78, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 39, 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]

(108−32, 108, 576)-Net in Base 8 — Constructive

(76, 108, 576)-net in base 8, using

8 times m-reduction [i] based on (76, 116, 576)-net in base 8, using
- trace code for nets [i] based on (18, 58, 288)-net in base 64, using
  - 5 times m-reduction [i] based on (18, 63, 288)-net in base 64, using
    - base change [i] based on digital (9, 54, 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]

(108−32, 108, 2500)-Net over F₈ — Digital

Digital (76, 108, 2500)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−32, 108, 1211419)-Net in Base 8 — Upper bound on s

There is no (76, 108, 1211420)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 175982 393929 186258 076541 902686 101793 802775 551503 165918 298860 975585 660956 074946 094991 528535 123335 > 8¹⁰⁸ [i]