MinT - Best Known (51, 75, s)-Nets in Base 8

Best Known (51, 75, s)-Nets in Base 8

(51, 75, 368)-Net over F₈ — Constructive and digital

Digital (51, 75, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. digital (38, 62, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 31, 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]

(51, 75, 520)-Net in Base 8 — Constructive

(51, 75, 520)-net in base 8, using

1 times m-reduction [i] based on (51, 76, 520)-net in base 8, using
- base change [i] based on digital (32, 57, 520)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (32, 58, 520)-net over F₁₆, using
    - trace code for nets [i] based on digital (3, 29, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(51, 75, 1198)-Net over F₈ — Digital

Digital (51, 75, 1198)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 75, 333093)-Net in Base 8 — Upper bound on s

There is no (51, 75, 333094)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 53 920524 212233 607484 850636 096224 715398 883039 779729 353282 920682 576724 > 8⁷⁵ [i]