MinT - Best Known (40, 40+129, s)-Nets in Base 8

Best Known (40, 40+129, s)-Nets in Base 8

(40, 40+129, 98)-Net over F₈ — Constructive and digital

Digital (40, 169, 98)-net over F₈, using

t-expansion [i] based on digital (37, 169, 98)-net over F₈, using
- net from sequence [i] based on digital (37, 97)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 37 and N(F) ≥ 98, using
    - F₅ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]

(40, 40+129, 129)-Net over F₈ — Digital

Digital (40, 169, 129)-net over F₈, using

t-expansion [i] based on digital (38, 169, 129)-net over F₈, using
- net from sequence [i] based on digital (38, 128)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 38 and N(F) ≥ 129, using
    - a curve from the manYPoints database [i]

(40, 40+129, 787)-Net in Base 8 — Upper bound on s

There is no (40, 169, 788)-net in base 8, because

1 times m-reduction [i] would yield (40, 168, 788)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 54 768246 983232 586975 907544 499039 059328 504487 355833 709871 619954 993689 117318 059428 108460 973777 951770 176544 018149 738334 678696 434881 717406 116395 206198 805828 > 8¹⁶⁸ [i]