MinT - Best Known (136−89, 136, s)-Nets in Base 8

Best Known (136−89, 136, s)-Nets in Base 8

(136−89, 136, 98)-Net over F₈ — Constructive and digital

Digital (47, 136, 98)-net over F₈, using

t-expansion [i] based on digital (37, 136, 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]

(136−89, 136, 144)-Net over F₈ — Digital

Digital (47, 136, 144)-net over F₈, using

t-expansion [i] based on digital (45, 136, 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]

(136−89, 136, 1426)-Net in Base 8 — Upper bound on s

There is no (47, 136, 1427)-net in base 8, because

1 times m-reduction [i] would yield (47, 135, 1427)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 83 323167 703763 576401 021981 736689 012172 874029 105088 718617 187887 017971 684751 595400 267856 172923 743421 078131 667782 611277 910928 > 8¹³⁵ [i]