MinT - Best Known (172−127, 172, s)-Nets in Base 8

Best Known (172−127, 172, s)-Nets in Base 8

(172−127, 172, 98)-Net over F₈ — Constructive and digital

Digital (45, 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]

(172−127, 172, 144)-Net over F₈ — Digital

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]

(172−127, 172, 941)-Net in Base 8 — Upper bound on s

There is no (45, 172, 942)-net in base 8, because

1 times m-reduction [i] would yield (45, 171, 942)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27129 883916 104445 471741 211261 262756 144366 667838 133217 717214 469970 708181 748054 277857 920562 470540 342906 751258 006359 891795 399873 726347 900738 880397 826337 328064 > 8¹⁷¹ [i]