MinT - Best Known (162−99, 162, s)-Nets in Base 8

Best Known (162−99, 162, s)-Nets in Base 8

(162−99, 162, 98)-Net over F₈ — Constructive and digital

Digital (63, 162, 98)-net over F₈, using

t-expansion [i] based on digital (37, 162, 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]

(162−99, 162, 144)-Net over F₈ — Digital

Digital (63, 162, 144)-net over F₈, using

t-expansion [i] based on digital (45, 162, 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]

(162−99, 162, 2501)-Net in Base 8 — Upper bound on s

There is no (63, 162, 2502)-net in base 8, because

1 times m-reduction [i] would yield (63, 161, 2502)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 25 284819 126600 950723 155209 413557 059753 330567 116176 507091 151542 290352 934288 125774 704027 114716 883241 584781 851629 997298 332232 882487 114129 147752 347916 > 8¹⁶¹ [i]