MinT - Best Known (109−33, 109, s)-Nets in Base 8

Best Known (109−33, 109, s)-Nets in Base 8

(109−33, 109, 402)-Net over F₈ — Constructive and digital

Digital (76, 109, 402)-net over F₈, using

1 times m-reduction [i] based on digital (76, 110, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 28, 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 (48, 82, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 41, 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]

(109−33, 109, 576)-Net in Base 8 — Constructive

(76, 109, 576)-net in base 8, using

7 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]

(109−33, 109, 2193)-Net over F₈ — Digital

Digital (76, 109, 2193)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−33, 109, 1211419)-Net in Base 8 — Upper bound on s

There is no (76, 109, 1211420)-net in base 8, because

1 times m-reduction [i] would yield (76, 108, 1211420)-net in base 8, but
- 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]