MinT - Best Known (158−109, 158, s)-Nets in Base 8

Best Known (158−109, 158, s)-Nets in Base 8

(158−109, 158, 98)-Net over F₈ — Constructive and digital

Digital (49, 158, 98)-net over F₈, using

t-expansion [i] based on digital (37, 158, 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]

(158−109, 158, 144)-Net over F₈ — Digital

Digital (49, 158, 144)-net over F₈, using

t-expansion [i] based on digital (45, 158, 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]

(158−109, 158, 1231)-Net in Base 8 — Upper bound on s

There is no (49, 158, 1232)-net in base 8, because

1 times m-reduction [i] would yield (49, 157, 1232)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 6342 032438 405846 795311 899817 952176 753444 256368 997649 336927 054726 370704 655161 081754 567532 203500 854844 968939 180830 363390 136470 563661 833310 564252 > 8¹⁵⁷ [i]