MinT - Best Known (109−97, 109, s)-Nets in Base 8

Best Known (109−97, 109, s)-Nets in Base 8

(109−97, 109, 48)-Net over F₈ — Constructive and digital

Digital (12, 109, 48)-net over F₈, using

t-expansion [i] based on digital (11, 109, 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]

(109−97, 109, 49)-Net over F₈ — Digital

Digital (12, 109, 49)-net over F₈, using

net from sequence [i] based on digital (12, 48)-sequence over F₈, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 12 and N(F) ≥ 49, using
  - a curve from the manYPoints database [i]

(109−97, 109, 102)-Net in Base 8 — Upper bound on s

There is no (12, 109, 103)-net in base 8, because

19 times m-reduction [i] would yield (12, 90, 103)-net in base 8, but
- extracting embedded orthogonal array [i] would yield OA(8⁹⁰, 103, S₈, 78), but
  - the linear programming bound shows that MÂ â‰¥ 1021 576153 078735 923643 496828 494105 918576 721926 818258 715769 540997 787100 910918 672673 951195 332608 / 431023 326525 > 8⁹⁰ [i]