MinT - Best Known (22, 22+56, s)-Nets in Base 27

Best Known (22, 22+56, s)-Nets in Base 27

(22, 22+56, 112)-Net over F₂₇ — Constructive and digital

Digital (22, 78, 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+56, 116)-Net in Base 27 — Constructive

(22, 78, 116)-net in base 27, using

2 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+56, 163)-Net over F₂₇ — Digital

Digital (22, 78, 163)-net over F₂₇, using

t-expansion [i] based on digital (21, 78, 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+56, 4206)-Net in Base 27 — Upper bound on s

There is no (22, 78, 4207)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 4450 124700 409826 863336 221750 618481 113109 796194 048137 309052 320855 231617 547359 194268 304377 015118 507750 212496 093225 > 27⁷⁸ [i]