MinT - Best Known (169−115, 169, s)-Nets in Base 8

Best Known (169−115, 169, s)-Nets in Base 8

(169−115, 169, 98)-Net over F₈ — Constructive and digital

Digital (54, 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]

(169−115, 169, 144)-Net over F₈ — Digital

Digital (54, 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]

(169−115, 169, 1411)-Net in Base 8 — Upper bound on s

There is no (54, 169, 1412)-net in base 8, because

1 times m-reduction [i] would yield (54, 168, 1412)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 53 023598 678639 594951 863943 938217 440163 527797 913889 554936 259051 373951 838252 300168 702184 163215 213266 838503 289315 223417 990875 974006 474186 242714 908387 495792 > 8¹⁶⁸ [i]