MinT - Best Known (170−144, 170, s)-Nets in Base 8

Best Known (170−144, 170, s)-Nets in Base 8

(170−144, 170, 65)-Net over F₈ — Constructive and digital

Digital (26, 170, 65)-net over F₈, using

t-expansion [i] based on digital (14, 170, 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]

(170−144, 170, 86)-Net over F₈ — Digital

Digital (26, 170, 86)-net over F₈, using

t-expansion [i] based on digital (25, 170, 86)-net over F₈, using
- net from sequence [i] based on digital (25, 85)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 25 and N(F) ≥ 86, using
    - a curve from the manYPoints database [i]

(170−144, 170, 388)-Net over F₈ — Upper bound on s (digital)

There is no digital (26, 170, 389)-net over F₈, because

extracting embedded orthogonal array [i] would yield linear OA(8¹⁷⁰, 389, F₈, 144) (dual of [389, 219, 145]-code), but
- residual code [i] would yield OA(8²⁶, 244, S₈, 18), but
  - the linear programming bound shows that MÂ â‰¥ 377080 730326 957484 118808 951055 960692 370282 905600 / 1 205370 411814 620505 759077 > 8²⁶ [i]

(170−144, 170, 484)-Net in Base 8 — Upper bound on s

There is no (26, 170, 485)-net in base 8, because

20 times m-reduction [i] would yield (26, 150, 485)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2954 148216 718685 416735 644241 256412 783238 023580 258894 240012 823558 102570 212059 174048 081606 966219 721886 096105 577414 716791 575450 240152 993696 > 8¹⁵⁰ [i]