MinT - Best Known (117−65, 117, s)-Nets in Base 8

Best Known (117−65, 117, s)-Nets in Base 8

(117−65, 117, 98)-Net over F₈ — Constructive and digital

Digital (52, 117, 98)-net over F₈, using

t-expansion [i] based on digital (37, 117, 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]

(117−65, 117, 144)-Net over F₈ — Digital

Digital (52, 117, 144)-net over F₈, using

t-expansion [i] based on digital (45, 117, 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]

(117−65, 117, 3411)-Net in Base 8 — Upper bound on s

There is no (52, 117, 3412)-net in base 8, because

1 times m-reduction [i] would yield (52, 116, 3412)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 576 264976 099288 141290 603118 720799 290043 328999 069273 549728 199608 490760 001904 526360 394727 445539 601770 466810 > 8¹¹⁶ [i]