MinT - Best Known (67, 67+91, s)-Nets in Base 8

Best Known (67, 67+91, s)-Nets in Base 8

(67, 67+91, 100)-Net over F₈ — Constructive and digital

Digital (67, 158, 100)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (8, 53, 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, 105, 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]

(67, 67+91, 145)-Net over F₈ — Digital

Digital (67, 158, 145)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(67, 67+91, 156)-Net in Base 8

(67, 158, 156)-net in base 8, using

2 times m-reduction [i] based on (67, 160, 156)-net in base 8, using
- base change [i] based on digital (27, 120, 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]

(67, 67+91, 3534)-Net in Base 8 — Upper bound on s

There is no (67, 158, 3535)-net in base 8, because

1 times m-reduction [i] would yield (67, 157, 3535)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6099 373766 141581 628416 255058 502264 995675 708627 598495 761588 841357 719366 348532 819600 668253 957816 204256 606246 286504 438903 011264 122478 049633 638260 > 8¹⁵⁷ [i]