MinT - Best Known (172−115, 172, s)-Nets in Base 8

Best Known (172−115, 172, s)-Nets in Base 8

(172−115, 172, 98)-Net over F₈ — Constructive and digital

Digital (57, 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]

(172−115, 172, 144)-Net over F₈ — Digital

Digital (57, 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]

(172−115, 172, 1579)-Net in Base 8 — Upper bound on s

There is no (57, 172, 1580)-net in base 8, because

1 times m-reduction [i] would yield (57, 171, 1580)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27699 537102 101063 320866 646403 158068 631465 609230 110373 045195 062893 453978 325249 683893 830345 198246 904377 851780 176434 526084 188767 192093 519416 276539 985552 122016 > 8¹⁷¹ [i]