MinT - Best Known (69, 69+93, s)-Nets in Base 8

Best Known (69, 69+93, s)-Nets in Base 8

(69, 69+93, 110)-Net over F₈ — Constructive and digital

Digital (69, 162, 110)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 55, 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]
2. digital (14, 107, 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+93, 150)-Net over F₈ — Digital

Digital (69, 162, 150)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+93, 156)-Net in Base 8

(69, 162, 156)-net in base 8, using

6 times m-reduction [i] based on (69, 168, 156)-net in base 8, using
- base change [i] based on digital (27, 126, 156)-net over F₁₆, using
  - net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
      - a curve from the manYPoints database [i]

(69, 69+93, 3694)-Net in Base 8 — Upper bound on s

There is no (69, 162, 3695)-net in base 8, because

1 times m-reduction [i] would yield (69, 161, 3695)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 141958 923549 239491 290552 964607 183415 025384 843936 111720 848586 323267 047180 878353 789950 699227 848013 754192 885448 009174 032508 220336 229627 788785 852071 > 8¹⁶¹ [i]