MinT - Best Known (57, 57+99, s)-Nets in Base 8

Best Known (57, 57+99, s)-Nets in Base 8

(57, 57+99, 98)-Net over F₈ — Constructive and digital

Digital (57, 156, 98)-net over F₈, using

t-expansion [i] based on digital (37, 156, 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]

(57, 57+99, 144)-Net over F₈ — Digital

Digital (57, 156, 144)-net over F₈, using

t-expansion [i] based on digital (45, 156, 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]

(57, 57+99, 1932)-Net in Base 8 — Upper bound on s

There is no (57, 156, 1933)-net in base 8, because

1 times m-reduction [i] would yield (57, 155, 1933)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 97 281749 171985 357522 807874 881944 076956 813381 501075 911757 572178 033391 249878 364134 135226 717116 345822 577885 067373 994760 127766 285424 929996 461536 > 8¹⁵⁵ [i]