MinT - Best Known (22, 22+53, s)-Nets in Base 27

Best Known (22, 22+53, s)-Nets in Base 27

(22, 22+53, 112)-Net over F₂₇ — Constructive and digital

Digital (22, 75, 112)-net over F₂₇, using

net from sequence [i] based on digital (22, 111)-sequence over F₂₇, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 22 and N(F) ≥ 112, using
  - an explicitly constructive algebraic function field [i]

(22, 22+53, 116)-Net in Base 27 — Constructive

(22, 75, 116)-net in base 27, using

5 times m-reduction [i] based on (22, 80, 116)-net in base 27, using
- base change [i] based on digital (2, 60, 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]

(22, 22+53, 163)-Net over F₂₇ — Digital

Digital (22, 75, 163)-net over F₂₇, using

t-expansion [i] based on digital (21, 75, 163)-net over F₂₇, using
- net from sequence [i] based on digital (21, 162)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 21 and N(F) ≥ 163, using
    - a curve from the manYPoints database [i]

(22, 22+53, 4797)-Net in Base 27 — Upper bound on s

There is no (22, 75, 4798)-net in base 27, because

1 times m-reduction [i] would yield (22, 74, 4798)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 8374 535458 728955 475715 429801 665243 553870 945944 068182 246244 897501 198425 186239 272216 267069 610319 247204 398957 > 27⁷⁴ [i]