MinT - Best Known (52−27, 52, s)-Nets in Base 8

Best Known (52−27, 52, s)-Nets in Base 8

(52−27, 52, 69)-Net over F₈ — Constructive and digital

Digital (25, 52, 69)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 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, 36, 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]

(52−27, 52, 91)-Net over F₈ — Digital

Digital (25, 52, 91)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(52−27, 52, 2818)-Net in Base 8 — Upper bound on s

There is no (25, 52, 2819)-net in base 8, because

1 times m-reduction [i] would yield (25, 51, 2819)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11453 295812 949764 613246 793811 747370 300568 586880 > 8⁵¹ [i]