MinT - Best Known (172−139, 172, s)-Nets in Base 8

Best Known (172−139, 172, s)-Nets in Base 8

(172−139, 172, 65)-Net over F₈ — Constructive and digital

Digital (33, 172, 65)-net over F₈, using

t-expansion [i] based on digital (14, 172, 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]

(172−139, 172, 97)-Net over F₈ — Digital

Digital (33, 172, 97)-net over F₈, using

t-expansion [i] based on digital (28, 172, 97)-net over F₈, using
- net from sequence [i] based on digital (28, 96)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 28 and N(F) ≥ 97, using
    - a curve from the manYPoints database [i]

(172−139, 172, 612)-Net in Base 8 — Upper bound on s

There is no (33, 172, 613)-net in base 8, because

1 times m-reduction [i] would yield (33, 171, 613)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27835 297954 585646 858983 149595 781152 895101 877401 163227 972746 073397 953628 908569 956167 617394 341624 366594 604127 693427 109529 321241 202339 776156 200410 378464 126584 > 8¹⁷¹ [i]