MinT - Best Known (51, 51+88, s)-Nets in Base 8

Best Known (51, 51+88, s)-Nets in Base 8

(51, 51+88, 98)-Net over F₈ — Constructive and digital

Digital (51, 139, 98)-net over F₈, using

t-expansion [i] based on digital (37, 139, 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+88, 144)-Net over F₈ — Digital

Digital (51, 139, 144)-net over F₈, using

t-expansion [i] based on digital (45, 139, 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+88, 1729)-Net in Base 8 — Upper bound on s

There is no (51, 139, 1730)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 344168 425242 781999 133105 358320 830078 319592 465788 875145 703301 785977 144037 145296 457187 264253 054044 417087 414166 955499 876122 659792 > 8¹³⁹ [i]