MinT - Best Known (172−121, 172, s)-Nets in Base 8

Best Known (172−121, 172, s)-Nets in Base 8

(172−121, 172, 98)-Net over F₈ — Constructive and digital

Digital (51, 172, 98)-net over F₈, using

t-expansion [i] based on digital (37, 172, 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]

(172−121, 172, 144)-Net over F₈ — Digital

Digital (51, 172, 144)-net over F₈, using

t-expansion [i] based on digital (45, 172, 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]

(172−121, 172, 1204)-Net in Base 8 — Upper bound on s

There is no (51, 172, 1205)-net in base 8, because

1 times m-reduction [i] would yield (51, 171, 1205)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 28064 832316 080182 548996 622351 470319 210496 860888 135887 856563 287484 790794 746250 774481 675272 478629 378770 288671 073488 196587 023587 284595 968889 634372 891078 632688 > 8¹⁷¹ [i]