MinT - Best Known (165−95, 165, s)-Nets in Base 8

Best Known (165−95, 165, s)-Nets in Base 8

(165−95, 165, 110)-Net over F₈ — Constructive and digital

Digital (70, 165, 110)-net over F₈, using

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

(165−95, 165, 151)-Net over F₈ — Digital

Digital (70, 165, 151)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(165−95, 165, 156)-Net in Base 8

(70, 165, 156)-net in base 8, using

7 times m-reduction [i] based on (70, 172, 156)-net in base 8, using
- base change [i] based on digital (27, 129, 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]

(165−95, 165, 3687)-Net in Base 8 — Upper bound on s

There is no (70, 165, 3688)-net in base 8, because

1 times m-reduction [i] would yield (70, 164, 3688)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12868 194890 407634 139547 726711 850872 336310 336317 760558 660245 497073 335406 217862 320486 910087 197679 557044 125370 563722 800203 741057 133737 360107 998950 771336 > 8¹⁶⁴ [i]