MinT - Best Known (169−101, 169, s)-Nets in Base 8

Best Known (169−101, 169, s)-Nets in Base 8

(169−101, 169, 98)-Net over F₈ — Constructive and digital

Digital (68, 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]

(169−101, 169, 144)-Net over F₈ — Digital

Digital (68, 169, 144)-net over F₈, using

t-expansion [i] based on digital (45, 169, 144)-net over F₈, using
- net from sequence [i] based on digital (45, 143)-sequence over F₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 45 and N(F) ≥ 144, using
    - Drinfeld modules of rank 1 [i]

(169−101, 169, 2981)-Net in Base 8 — Upper bound on s

There is no (68, 169, 2982)-net in base 8, because

1 times m-reduction [i] would yield (68, 168, 2982)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 53 173144 721228 900371 038384 529453 352735 198167 651643 955762 699397 188454 760019 101553 895758 648577 263578 354246 603859 483062 757576 067608 457076 220994 377172 658416 > 8¹⁶⁸ [i]