MinT - Best Known (35, 82, s)-Nets in Base 8

Best Known (35, 82, s)-Nets in Base 8

(35, 82, 69)-Net over F₈ — Constructive and digital

Digital (35, 82, 69)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (9, 56, 45)-net over F₈, using
  - net from sequence [i] based on digital (9, 44)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 9 and N(F) ≥ 45, using
      - an explicitly constructive algebraic function field [i]

(35, 82, 112)-Net over F₈ — Digital

Digital (35, 82, 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]

(35, 82, 2026)-Net in Base 8 — Upper bound on s

There is no (35, 82, 2027)-net in base 8, because

1 times m-reduction [i] would yield (35, 81, 2027)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 14 203536 048275 945060 170944 583071 887579 400508 514877 892160 641401 663037 992496 > 8⁸¹ [i]