MinT - Best Known (52, 52+109, s)-Nets in Base 8

Best Known (52, 52+109, s)-Nets in Base 8

(52, 52+109, 98)-Net over F₈ — Constructive and digital

Digital (52, 161, 98)-net over F₈, using

t-expansion [i] based on digital (37, 161, 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]

(52, 52+109, 144)-Net over F₈ — Digital

Digital (52, 161, 144)-net over F₈, using

t-expansion [i] based on digital (45, 161, 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]

(52, 52+109, 1385)-Net in Base 8 — Upper bound on s

There is no (52, 161, 1386)-net in base 8, because

1 times m-reduction [i] would yield (52, 160, 1386)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3 126061 248544 741845 061234 550592 542151 599674 727700 363874 125925 851461 127038 991576 284849 559827 792091 936600 715594 186885 154660 031981 562528 844938 453744 > 8¹⁶⁰ [i]