MinT - Best Known (67, 172, s)-Nets in Base 8

Best Known (67, 172, s)-Nets in Base 8

(67, 172, 98)-Net over F₈ — Constructive and digital

Digital (67, 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]

(67, 172, 144)-Net over F₈ — Digital

Digital (67, 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]

(67, 172, 2662)-Net in Base 8 — Upper bound on s

There is no (67, 172, 2663)-net in base 8, because

1 times m-reduction [i] would yield (67, 171, 2663)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27181 167809 605160 186378 706191 810305 876931 618631 291174 347127 692502 299121 439025 449859 851461 190308 671269 054491 257403 583692 477110 504922 036150 063824 497647 124212 > 8¹⁷¹ [i]