MinT - Best Known (69, 69+89, s)-Nets in Base 8

Best Known (69, 69+89, s)-Nets in Base 8

(69, 69+89, 113)-Net over F₈ — Constructive and digital

Digital (69, 158, 113)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (11, 55, 48)-net over F₈, using
  - net from sequence [i] based on digital (11, 47)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 11 and N(F) ≥ 48, using
      - a function field by SÃ©mirat [i]
2. digital (14, 103, 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]

(69, 69+89, 158)-Net over F₈ — Digital

Digital (69, 158, 158)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+89, 161)-Net in Base 8

(69, 158, 161)-net in base 8, using

2 times m-reduction [i] based on (69, 160, 161)-net in base 8, using
- base change [i] based on digital (29, 120, 161)-net over F₁₆, using
  - net from sequence [i] based on digital (29, 160)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 29 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(69, 69+89, 4085)-Net in Base 8 — Upper bound on s

There is no (69, 158, 4086)-net in base 8, because

1 times m-reduction [i] would yield (69, 157, 4086)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6110 588267 055218 991614 115978 494038 650795 672908 556228 049287 393568 746007 617279 268307 859496 076378 472790 520481 729315 231415 059779 570798 855004 546834 > 8¹⁵⁷ [i]