MinT - Best Known (57, 57+101, s)-Nets in Base 8

Best Known (57, 57+101, s)-Nets in Base 8

(57, 57+101, 98)-Net over F₈ — Constructive and digital

Digital (57, 158, 98)-net over F₈, using

t-expansion [i] based on digital (37, 158, 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+101, 144)-Net over F₈ — Digital

Digital (57, 158, 144)-net over F₈, using

t-expansion [i] based on digital (45, 158, 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+101, 1875)-Net in Base 8 — Upper bound on s

There is no (57, 158, 1876)-net in base 8, because

1 times m-reduction [i] would yield (57, 157, 1876)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6235 602057 163855 276061 400078 275388 554267 816045 639248 249416 644651 341927 848444 774203 022061 844115 898835 229715 115785 593593 506793 588388 457738 153374 > 8¹⁵⁷ [i]