MinT - Best Known (135−75, 135, s)-Nets in Base 8

Best Known (135−75, 135, s)-Nets in Base 8

(135−75, 135, 110)-Net over F₈ — Constructive and digital

Digital (60, 135, 110)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (9, 46, 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, 89, 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]

(135−75, 135, 146)-Net over F₈ — Digital

Digital (60, 135, 146)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(135−75, 135, 150)-Net in Base 8

(60, 135, 150)-net in base 8, using

1 times m-reduction [i] based on (60, 136, 150)-net in base 8, using
- base change [i] based on digital (26, 102, 150)-net over F₁₆, using
  - net from sequence [i] based on digital (26, 149)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 26 and N(F) ≥ 150, using
      - a curve from the manYPoints database [i]

(135−75, 135, 3880)-Net in Base 8 — Upper bound on s

There is no (60, 135, 3881)-net in base 8, because

1 times m-reduction [i] would yield (60, 134, 3881)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 10 393342 167150 693772 294092 694755 205405 895647 987149 187531 479999 613588 800238 445207 626539 605496 323011 101679 286098 615366 900632 > 8¹³⁴ [i]