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

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

(57−29, 57, 74)-Net over F₈ — Constructive and digital

Digital (28, 57, 74)-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 (14, 43, 65)-net over F₈, using
  - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
      - the Suzuki function field over F₈ [i]

(57−29, 57, 106)-Net over F₈ — Digital

Digital (28, 57, 106)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(57−29, 57, 3529)-Net in Base 8 — Upper bound on s

There is no (28, 57, 3530)-net in base 8, because

1 times m-reduction [i] would yield (28, 56, 3530)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 375 337414 231560 616739 004355 216184 384627 000932 599904 > 8⁵⁶ [i]