MinT - Best Known (78−43, 78, s)-Nets in Base 8

Best Known (78−43, 78, s)-Nets in Base 8

(78−43, 78, 74)-Net over F₈ — Constructive and digital

Digital (35, 78, 74)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 21, 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, 57, 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]

(78−43, 78, 112)-Net over F₈ — Digital

Digital (35, 78, 112)-net over F₈, using

net from sequence [i] based on digital (35, 111)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 35 and N(F) ≥ 112, using
  - a curve from the manYPoints database [i]

(78−43, 78, 2526)-Net in Base 8 — Upper bound on s

There is no (35, 78, 2527)-net in base 8, because

1 times m-reduction [i] would yield (35, 77, 2527)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3468 556295 227774 314553 376043 899117 607622 202485 251670 021314 073103 713730 > 8⁷⁷ [i]