MinT - Best Known (51, 51+25, s)-Nets in Base 8

Best Known (51, 51+25, s)-Nets in Base 8

(51, 51+25, 363)-Net over F₈ — Constructive and digital

Digital (51, 76, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (39, 64, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 32, 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, 51+25, 520)-Net in Base 8 — Constructive

(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, 51+25, 1026)-Net over F₈ — Digital

Digital (51, 76, 1026)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(51, 51+25, 333093)-Net in Base 8 — Upper bound on s

There is no (51, 76, 333094)-net in base 8, because

1 times m-reduction [i] would yield (51, 75, 333094)-net in base 8, but
- 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]