MinT - Best Known (57, 57+29, s)-Nets in Base 8

Best Known (57, 57+29, s)-Nets in Base 8

(57, 57+29, 363)-Net over F₈ — Constructive and digital

Digital (57, 86, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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 (43, 72, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 36, 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]

(57, 57+29, 520)-Net in Base 8 — Constructive

(57, 86, 520)-net in base 8, using

trace code for nets [i] based on (14, 43, 260)-net in base 64, using
- 1 times m-reduction [i] based on (14, 44, 260)-net in base 64, using
  - base change [i] based on digital (3, 33, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(57, 57+29, 973)-Net over F₈ — Digital

Digital (57, 86, 973)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+29, 262658)-Net in Base 8 — Upper bound on s

There is no (57, 86, 262659)-net in base 8, because

1 times m-reduction [i] would yield (57, 85, 262659)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 57896 588564 172754 012638 551356 812476 957303 530471 188857 686727 894078 778202 730368 > 8⁸⁵ [i]