MinT - Best Known (163−117, 163, s)-Nets in Base 8

Best Known (163−117, 163, s)-Nets in Base 8

(163−117, 163, 98)-Net over F₈ — Constructive and digital

Digital (46, 163, 98)-net over F₈, using

t-expansion [i] based on digital (37, 163, 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]

(163−117, 163, 144)-Net over F₈ — Digital

Digital (46, 163, 144)-net over F₈, using

t-expansion [i] based on digital (45, 163, 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]

(163−117, 163, 1031)-Net in Base 8 — Upper bound on s

There is no (46, 163, 1032)-net in base 8, because

1 times m-reduction [i] would yield (46, 162, 1032)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 204 765027 977599 820538 677213 282895 652554 131429 858424 146389 634276 869344 024674 687867 814337 906613 840038 213839 639445 508989 468510 360392 902052 576992 718720 > 8¹⁶² [i]