MinT - Best Known (137−42, 137, s)-Nets in Base 8

Best Known (137−42, 137, s)-Nets in Base 8

(137−42, 137, 419)-Net over F₈ — Constructive and digital

Digital (95, 137, 419)-net over F₈, using

3 times m-reduction [i] based on digital (95, 140, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 36, 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 (59, 104, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 52, 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]

(137−42, 137, 576)-Net in Base 8 — Constructive

(95, 137, 576)-net in base 8, using

13 times m-reduction [i] based on (95, 150, 576)-net in base 8, using
- trace code for nets [i] based on (20, 75, 288)-net in base 64, using
  - 2 times m-reduction [i] based on (20, 77, 288)-net in base 64, using
    - base change [i] based on digital (9, 66, 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]

(137−42, 137, 2422)-Net over F₈ — Digital

Digital (95, 137, 2422)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−42, 137, 965996)-Net in Base 8 — Upper bound on s

There is no (95, 137, 965997)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 5288 457204 117891 337232 379546 172306 621284 524540 805648 813067 883950 131854 461127 996666 073408 387184 591004 768160 489994 764478 550240 > 8¹³⁷ [i]