MinT - Best Known (172−113, 172, s)-Nets in Base 8

Best Known (172−113, 172, s)-Nets in Base 8

(172−113, 172, 98)-Net over F₈ — Constructive and digital

Digital (59, 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]

(172−113, 172, 144)-Net over F₈ — Digital

Digital (59, 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]

(172−113, 172, 1739)-Net in Base 8 — Upper bound on s

There is no (59, 172, 1740)-net in base 8, because

1 times m-reduction [i] would yield (59, 171, 1740)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27080 470041 109076 211544 334443 622864 630786 196997 229836 600041 417290 606435 369936 796232 464817 788518 542645 621078 471957 082735 271466 433837 804961 320820 008934 503132 > 8¹⁷¹ [i]