MinT - Best Known (12, 12+17, s)-Nets in Base 81

Best Known (12, 12+17, s)-Nets in Base 81

(12, 12+17, 232)-Net over F₈₁ — Constructive and digital

Digital (12, 29, 232)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (2, 10, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 2 and N(F) ≥ 116, using
      - a function field by SÃ©mirat [i]
2. digital (2, 19, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁ (see above)

(12, 12+17, 298)-Net over F₈₁ — Digital

Digital (12, 29, 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]

(12, 12+17, 225056)-Net in Base 81 — Upper bound on s

There is no (12, 29, 225057)-net in base 81, because

1 times m-reduction [i] would yield (12, 28, 225057)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 273901 966953 989113 934875 394314 004094 559792 603795 902081 > 81²⁸ [i]