MinT - Best Known (162−107, 162, s)-Nets in Base 8

Best Known (162−107, 162, s)-Nets in Base 8

(162−107, 162, 98)-Net over F₈ — Constructive and digital

Digital (55, 162, 98)-net over F₈, using

t-expansion [i] based on digital (37, 162, 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]

(162−107, 162, 144)-Net over F₈ — Digital

Digital (55, 162, 144)-net over F₈, using

t-expansion [i] based on digital (45, 162, 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]

(162−107, 162, 1596)-Net in Base 8 — Upper bound on s

There is no (55, 162, 1597)-net in base 8, because

1 times m-reduction [i] would yield (55, 161, 1597)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 724516 642770 307481 795256 732838 045416 228161 697510 015871 146202 754640 706699 050114 995908 422215 897427 977724 079523 414324 137050 154867 660563 435516 147712 > 8¹⁶¹ [i]