MinT - Best Known (53−39, 53, s)-Nets in Base 81

Best Known (53−39, 53, s)-Nets in Base 81

(53−39, 53, 224)-Net over F₈₁ — Constructive and digital

Digital (14, 53, 224)-net over F₈₁, using

t-expansion [i] based on digital (13, 53, 224)-net over F₈₁, using
- net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
    - a function field by SÃ©mirat [i]

(53−39, 53, 298)-Net over F₈₁ — Digital

Digital (14, 53, 298)-net over F₈₁, using

t-expansion [i] based on digital (12, 53, 298)-net over F₈₁, using
- net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
    - a curve from the manYPoints database [i]

(53−39, 53, 16561)-Net in Base 81 — Upper bound on s

There is no (14, 53, 16562)-net in base 81, because

1 times m-reduction [i] would yield (14, 52, 16562)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 1743 245320 470373 404268 348598 091770 366829 697278 678911 188207 441452 878013 275724 003469 047072 092889 129441 > 81⁵² [i]