MinT - Best Known (12, ∞, s)-Nets in Base 8

Best Known (12, ∞, s)-Nets in Base 8

(12, ∞, 48)-Net over F₈ — Constructive and digital

Digital (12, m, 48)-net over F₈ for arbitrarily large m, using

net from sequence [i] based on digital (12, 47)-sequence over F₈, using
- t-expansion [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]

(12, ∞, 49)-Net over F₈ — Digital

Digital (12, m, 49)-net over F₈ for arbitrarily large m, 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]

(12, ∞, 102)-Net in Base 8 — Upper bound on s

There is no (12, m, 103)-net in base 8 for arbitrarily large m, because

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]