MinT - Best Known (8, 67, s)-Nets in Base 81

Best Known (8, 67, s)-Nets in Base 81

(8, 67, 172)-Net over F₈₁ — Constructive and digital

Digital (8, 67, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 67, 172)-net over F₈₁, using
- net from sequence [i] based on digital (7, 171)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 7 and N(F) ≥ 172, using
    - a function field by SÃ©mirat [i]

(8, 67, 226)-Net over F₈₁ — Digital

Digital (8, 67, 226)-net over F₈₁, using

net from sequence [i] based on digital (8, 225)-sequence over F₈₁, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 8 and N(F) ≥ 226, using
  - a curve from the manYPoints database [i]

(8, 67, 3202)-Net in Base 81 — Upper bound on s

There is no (8, 67, 3203)-net in base 81, because

1 times m-reduction [i] would yield (8, 66, 3203)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 913107 386138 130936 445073 332436 800916 777950 683729 734084 714168 604604 769857 047093 178549 004630 693849 897054 166477 971100 678259 181361 > 81⁶⁶ [i]