MinT - Best Known (126−39, 126, s)-Nets in Base 8

Best Known (126−39, 126, s)-Nets in Base 8

(126−39, 126, 419)-Net over F₈ — Constructive and digital

Digital (87, 126, 419)-net over F₈, using

8¹ times duplication [i] based on digital (86, 125, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 33, 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 (53, 92, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 46, 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]

(126−39, 126, 576)-Net in Base 8 — Constructive

(87, 126, 576)-net in base 8, using

10 times m-reduction [i] based on (87, 136, 576)-net in base 8, using
- trace code for nets [i] based on (19, 68, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(126−39, 126, 2138)-Net over F₈ — Digital

Digital (87, 126, 2138)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(126−39, 126, 989677)-Net in Base 8 — Upper bound on s

There is no (87, 126, 989678)-net in base 8, because

1 times m-reduction [i] would yield (87, 125, 989678)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 76957 369834 794354 291189 402102 846198 848151 068907 332770 832081 359271 263891 917274 616525 976221 106525 725666 386951 290588 > 8¹²⁵ [i]