MinT - Best Known (95, 95+51, s)-Nets in Base 8

Best Known (95, 95+51, s)-Nets in Base 8

(95, 95+51, 382)-Net over F₈ — Constructive and digital

Digital (95, 146, 382)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (5, 30, 28)-net over F₈, using
  - net from sequence [i] based on digital (5, 27)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 5 and N(F) ≥ 28, using
      - a function field by SÃ©mirat [i]
2. digital (65, 116, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 58, 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]

(95, 95+51, 576)-Net in Base 8 — Constructive

(95, 146, 576)-net in base 8, using

4 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]

(95, 95+51, 1232)-Net over F₈ — Digital

Digital (95, 146, 1232)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(95, 95+51, 251433)-Net in Base 8 — Upper bound on s

There is no (95, 146, 251434)-net in base 8, because

1 times m-reduction [i] would yield (95, 145, 251434)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 88731 495702 142557 600140 564535 189809 023402 643333 368503 114445 220454 244487 276417 004923 838567 181814 027068 243698 782011 027909 775896 625422 > 8¹⁴⁵ [i]