MinT - Best Known (8, 8+57, s)-Nets in Base 81

Best Known (8, 8+57, s)-Nets in Base 81

(8, 8+57, 172)-Net over F₈₁ — Constructive and digital

Digital (8, 65, 172)-net over F₈₁, using

t-expansion [i] based on digital (7, 65, 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, 8+57, 226)-Net over F₈₁ — Digital

Digital (8, 65, 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, 8+57, 3238)-Net in Base 81 — Upper bound on s

There is no (8, 65, 3239)-net in base 81, because

1 times m-reduction [i] would yield (8, 64, 3239)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 139 871383 396789 547428 431587 412530 555603 498002 594439 535379 943591 133917 267374 377083 563205 313465 900677 756888 786394 585036 631361 > 81⁶⁴ [i]