MinT - Best Known (171−113, 171, s)-Nets in Base 8

Best Known (171−113, 171, s)-Nets in Base 8

(171−113, 171, 98)-Net over F₈ — Constructive and digital

Digital (58, 171, 98)-net over F₈, using

t-expansion [i] based on digital (37, 171, 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]

(171−113, 171, 144)-Net over F₈ — Digital

Digital (58, 171, 144)-net over F₈, using

t-expansion [i] based on digital (45, 171, 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]

(171−113, 171, 1674)-Net in Base 8 — Upper bound on s

There is no (58, 171, 1675)-net in base 8, because

1 times m-reduction [i] would yield (58, 170, 1675)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 3352 472864 383807 104944 613816 621956 093593 607203 755495 794382 096045 263445 041692 132791 352894 546497 164945 203634 546445 774701 942336 707357 371360 035408 256632 754136 > 8¹⁷⁰ [i]