MinT - Best Known (135−42, 135, s)-Nets in Base 8

Best Known (135−42, 135, s)-Nets in Base 8

(135−42, 135, 419)-Net over F₈ — Constructive and digital

Digital (93, 135, 419)-net over F₈, using

t-expansion [i] based on digital (92, 135, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 35, 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 (57, 100, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 50, 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]

(135−42, 135, 576)-Net in Base 8 — Constructive

(93, 135, 576)-net in base 8, using

11 times m-reduction [i] based on (93, 146, 576)-net in base 8, using
- trace code for nets [i] based on (20, 73, 288)-net in base 64, using
  - 4 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]

(135−42, 135, 2190)-Net over F₈ — Digital

Digital (93, 135, 2190)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−42, 135, 792439)-Net in Base 8 — Upper bound on s

There is no (93, 135, 792440)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 633989 186126 519511 773468 226708 907564 431936 853792 572340 010774 127156 519217 564546 175398 993461 088554 949090 851951 025451 707849 > 8¹³⁵ [i]