MinT - Best Known (21, 21+53, s)-Nets in Base 27

Best Known (21, 21+53, s)-Nets in Base 27

(21, 21+53, 108)-Net over F₂₇ — Constructive and digital

Digital (21, 74, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 74, 108)-net over F₂₇, using
- net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
    - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

(21, 21+53, 116)-Net in Base 27 — Constructive

(21, 74, 116)-net in base 27, using

2 times m-reduction [i] based on (21, 76, 116)-net in base 27, using
- base change [i] based on digital (2, 57, 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]

(21, 21+53, 163)-Net over F₂₇ — Digital

Digital (21, 74, 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]

(21, 21+53, 4224)-Net in Base 27 — Upper bound on s

There is no (21, 74, 4225)-net in base 27, because

1 times m-reduction [i] would yield (21, 73, 4225)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 309 869673 875070 203011 636485 798885 767576 790976 652965 588149 525244 118499 599948 494572 620240 245483 655393 594745 > 27⁷³ [i]