MinT - Best Known (52, 52+99, s)-Nets in Base 8

Best Known (52, 52+99, s)-Nets in Base 8

(52, 52+99, 98)-Net over F₈ — Constructive and digital

Digital (52, 151, 98)-net over F₈, using

t-expansion [i] based on digital (37, 151, 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]

(52, 52+99, 144)-Net over F₈ — Digital

Digital (52, 151, 144)-net over F₈, using

t-expansion [i] based on digital (45, 151, 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]

(52, 52+99, 1556)-Net in Base 8 — Upper bound on s

There is no (52, 151, 1557)-net in base 8, because

1 times m-reduction [i] would yield (52, 150, 1557)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2916 951154 742946 188939 684536 999641 536184 067552 791996 313572 092913 671729 406725 710456 552538 116897 092258 443776 676333 195935 316398 275423 045568 > 8¹⁵⁰ [i]