MinT - Best Known (131−38, 131, s)-Nets in Base 8

Best Known (131−38, 131, s)-Nets in Base 8

(131−38, 131, 514)-Net over F₈ — Constructive and digital

Digital (93, 131, 514)-net over F₈, using

1 times m-reduction [i] based on digital (93, 132, 514)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (21, 40, 160)-net over F₈, using
    - trace code for nets [i] based on digital (1, 20, 80)-net over F₆₄, using
      - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [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]

(131−38, 131, 593)-Net in Base 8 — Constructive

(93, 131, 593)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (2, 21, 17)-net over F₈, using
  - net from sequence [i] based on digital (2, 16)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 2 and N(F) ≥ 17, using
      - a function field by SÃ©mirat [i]
2. (72, 110, 576)-net in base 8, using
  - trace code for nets [i] based on (17, 55, 288)-net in base 64, using
    - 1 times m-reduction [i] based on (17, 56, 288)-net in base 64, using
      - base change [i] based on digital (9, 48, 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]

(131−38, 131, 3316)-Net over F₈ — Digital

Digital (93, 131, 3316)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(131−38, 131, 1908457)-Net in Base 8 — Upper bound on s

There is no (93, 131, 1908458)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 20173 828101 917284 015189 914757 887692 893652 654037 958430 105148 834339 428514 412877 696664 655575 557740 956501 809242 527120 241464 > 8¹³¹ [i]