MinT - Best Known (133−43, 133, s)-Nets in Base 8

Best Known (133−43, 133, s)-Nets in Base 8

(133−43, 133, 402)-Net over F₈ — Constructive and digital

Digital (90, 133, 402)-net over F₈, using

8¹ times duplication [i] based on digital (89, 132, 402)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (11, 32, 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 (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]

(133−43, 133, 576)-Net in Base 8 — Constructive

(90, 133, 576)-net in base 8, using

t-expansion [i] based on (89, 133, 576)-net in base 8, using
- 7 times m-reduction [i] based on (89, 140, 576)-net in base 8, using
  - trace code for nets [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]

(133−43, 133, 1729)-Net over F₈ — Digital

Digital (90, 133, 1729)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(133−43, 133, 588776)-Net in Base 8 — Upper bound on s

There is no (90, 133, 588777)-net in base 8, because

1 times m-reduction [i] would yield (90, 132, 588777)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 161392 689722 597152 276441 296107 663862 028752 554747 614223 719322 468706 680092 790467 748514 950647 311746 109841 119097 243087 122480 > 8¹³² [i]