MinT - Best Known (137−87, 137, s)-Nets in Base 8

Best Known (137−87, 137, s)-Nets in Base 8

(137−87, 137, 98)-Net over F₈ — Constructive and digital

Digital (50, 137, 98)-net over F₈, using

t-expansion [i] based on digital (37, 137, 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]

(137−87, 137, 144)-Net over F₈ — Digital

Digital (50, 137, 144)-net over F₈, using

t-expansion [i] based on digital (45, 137, 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]

(137−87, 137, 1705)-Net in Base 8 — Upper bound on s

There is no (50, 137, 1706)-net in base 8, because

1 times m-reduction [i] would yield (50, 136, 1706)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 670 749454 073136 406554 400073 984788 560577 941210 588433 585817 726303 889249 105351 242868 737033 505930 207643 770268 706735 985022 503168 > 8¹³⁶ [i]