MinT - Best Known (13, 13+19, s)-Nets in Base 81

Best Known (13, 13+19, s)-Nets in Base 81

(13, 13+19, 232)-Net over F₈₁ — Constructive and digital

Digital (13, 32, 232)-net over F₈₁, using

(u,Â u+v)-construction [i] based on
1. digital (2, 11, 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, 21, 116)-net over F₈₁, using
  - net from sequence [i] based on digital (2, 115)-sequence over F₈₁ (see above)

(13, 13+19, 298)-Net over F₈₁ — Digital

Digital (13, 32, 298)-net over F₈₁, using

t-expansion [i] based on digital (12, 32, 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]

(13, 13+19, 194232)-Net in Base 81 — Upper bound on s

There is no (13, 32, 194233)-net in base 81, because

1 times m-reduction [i] would yield (13, 31, 194233)-net in base 81, but
- the generalized Rao bound for nets shows that 81^mÂ â‰¥ 145558 231120 315004 947148 362466 623876 071454 561149 041495 342161 > 81³¹ [i]