MinT - Best Known (126−73, 126, s)-Nets in Base 8

Best Known (126−73, 126, s)-Nets in Base 8

(126−73, 126, 98)-Net over F₈ — Constructive and digital

Digital (53, 126, 98)-net over F₈, using

t-expansion [i] based on digital (37, 126, 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]

(126−73, 126, 144)-Net over F₈ — Digital

Digital (53, 126, 144)-net over F₈, using

t-expansion [i] based on digital (45, 126, 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]

(126−73, 126, 2765)-Net in Base 8 — Upper bound on s

There is no (53, 126, 2766)-net in base 8, because

1 times m-reduction [i] would yield (53, 125, 2766)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 77059 181225 284260 176227 152779 362460 505424 345264 816239 525977 331705 512007 524360 777616 539122 203464 493853 985031 533598 > 8¹²⁵ [i]