MinT - Best Known (165−117, 165, s)-Nets in Base 8

Best Known (165−117, 165, s)-Nets in Base 8

(165−117, 165, 98)-Net over F₈ — Constructive and digital

Digital (48, 165, 98)-net over F₈, using

t-expansion [i] based on digital (37, 165, 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]

(165−117, 165, 144)-Net over F₈ — Digital

Digital (48, 165, 144)-net over F₈, using

t-expansion [i] based on digital (45, 165, 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]

(165−117, 165, 1110)-Net in Base 8 — Upper bound on s

There is no (48, 165, 1111)-net in base 8, because

1 times m-reduction [i] would yield (48, 164, 1111)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 12844 111081 896179 650448 591832 672798 361667 359510 882393 130599 950380 611450 182860 788480 062500 483679 600561 260170 067972 238917 561546 300496 264449 664623 945544 > 8¹⁶⁴ [i]