MinT - Best Known (51, 51+83, s)-Nets in Base 8

Best Known (51, 51+83, s)-Nets in Base 8

(51, 51+83, 98)-Net over F₈ — Constructive and digital

Digital (51, 134, 98)-net over F₈, using

t-expansion [i] based on digital (37, 134, 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+83, 144)-Net over F₈ — Digital

Digital (51, 134, 144)-net over F₈, using

t-expansion [i] based on digital (45, 134, 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+83, 1934)-Net in Base 8 — Upper bound on s

There is no (51, 134, 1935)-net in base 8, because

1 times m-reduction [i] would yield (51, 133, 1935)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1 299977 516470 589726 880253 703060 255504 200796 707918 502005 100878 661697 003000 487506 447733 058543 681568 205630 179264 520150 927032 > 8¹³³ [i]