MinT - Best Known (125−40, 125, s)-Nets in Base 8

Best Known (125−40, 125, s)-Nets in Base 8

(125−40, 125, 402)-Net over F₈ — Constructive and digital

Digital (85, 125, 402)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 31, 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 (54, 94, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 47, 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]

(125−40, 125, 576)-Net in Base 8 — Constructive

(85, 125, 576)-net in base 8, using

7 times m-reduction [i] based on (85, 132, 576)-net in base 8, using
- trace code for nets [i] based on (19, 66, 288)-net in base 64, using
  - 4 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]

(125−40, 125, 1745)-Net over F₈ — Digital

Digital (85, 125, 1745)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(125−40, 125, 523010)-Net in Base 8 — Upper bound on s

There is no (85, 125, 523011)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 76958 724610 144310 514769 458949 954419 510756 456332 831621 846857 654682 927389 983416 548028 342108 215238 927190 456123 077824 > 8¹²⁵ [i]