MinT - Best Known (51, 51+85, s)-Nets in Base 8

Best Known (51, 51+85, s)-Nets in Base 8

(51, 51+85, 98)-Net over F₈ — Constructive and digital

Digital (51, 136, 98)-net over F₈, using

t-expansion [i] based on digital (37, 136, 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]

(51, 51+85, 144)-Net over F₈ — Digital

Digital (51, 136, 144)-net over F₈, using

t-expansion [i] based on digital (45, 136, 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]

(51, 51+85, 1859)-Net in Base 8 — Upper bound on s

There is no (51, 136, 1860)-net in base 8, because

1 times m-reduction [i] would yield (51, 135, 1860)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 83 740368 177393 640872 718510 557941 278117 609127 487316 032965 618122 286795 227732 416670 026063 678426 720285 118556 960145 761670 565976 > 8¹³⁵ [i]