MinT - Best Known (69, 69+95, s)-Nets in Base 8

Best Known (69, 69+95, s)-Nets in Base 8

(69, 69+95, 100)-Net over F₈ — Constructive and digital

Digital (69, 164, 100)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 55, 35)-net over F₈, using
  - net from sequence [i] based on digital (8, 34)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ III based on the algebraic function field F/F₈ with g(F)Â =Â 7, N(F)Â =Â 34, and 1 place with degree 2 [i] based on function field F/F₈ with g(F) = 7 and N(F) ≥ 34, using a function field by SÃ©mirat [i]
2. digital (14, 109, 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+95, 146)-Net over F₈ — Digital

Digital (69, 164, 146)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(69, 69+95, 156)-Net in Base 8

(69, 164, 156)-net in base 8, using

4 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+95, 3526)-Net in Base 8 — Upper bound on s

There is no (69, 164, 3527)-net in base 8, because

1 times m-reduction [i] would yield (69, 163, 3527)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1606 240796 471279 834472 290373 515596 983981 639638 584410 672764 888976 446925 222330 937443 145886 287921 516199 177891 152052 948488 322245 142592 643303 877860 552320 > 8¹⁶³ [i]