MinT - Best Known (136−83, 136, s)-Nets in Base 8

Best Known (136−83, 136, s)-Nets in Base 8

(136−83, 136, 98)-Net over F₈ — Constructive and digital

Digital (53, 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]

(136−83, 136, 144)-Net over F₈ — Digital

Digital (53, 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]

(136−83, 136, 2143)-Net in Base 8 — Upper bound on s

There is no (53, 136, 2144)-net in base 8, because

1 times m-reduction [i] would yield (53, 135, 2144)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 82 699010 652007 958976 166828 608692 611142 551609 000310 828139 699056 497850 261259 639753 722901 941689 456999 006288 917700 388524 311810 > 8¹³⁵ [i]