MinT - Best Known (167−121, 167, s)-Nets in Base 8

Best Known (167−121, 167, s)-Nets in Base 8

(167−121, 167, 98)-Net over F₈ — Constructive and digital

Digital (46, 167, 98)-net over F₈, using

t-expansion [i] based on digital (37, 167, 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]

(167−121, 167, 144)-Net over F₈ — Digital

Digital (46, 167, 144)-net over F₈, using

t-expansion [i] based on digital (45, 167, 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]

(167−121, 167, 1006)-Net in Base 8 — Upper bound on s

There is no (46, 167, 1007)-net in base 8, because

1 times m-reduction [i] would yield (46, 166, 1007)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 839742 902061 543490 216671 362569 997751 638787 225549 089133 200895 949319 580726 142901 896923 111922 028689 977900 180341 869262 863500 590835 414829 008081 975317 417204 > 8¹⁶⁶ [i]