MinT - Best Known (139−87, 139, s)-Nets in Base 8

Best Known (139−87, 139, s)-Nets in Base 8

(139−87, 139, 98)-Net over F₈ — Constructive and digital

Digital (52, 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]

(139−87, 139, 144)-Net over F₈ — Digital

Digital (52, 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]

(139−87, 139, 1881)-Net in Base 8 — Upper bound on s

There is no (52, 139, 1882)-net in base 8, because

1 times m-reduction [i] would yield (52, 138, 1882)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 42951 495045 548444 799314 246891 559925 255852 808712 831181 821316 706022 252017 792598 882129 158025 027427 676635 620718 320907 029660 832048 > 8¹³⁸ [i]