MinT - Best Known (21, 21+49, s)-Nets in Base 27

Best Known (21, 21+49, s)-Nets in Base 27

(21, 21+49, 108)-Net over F₂₇ — Constructive and digital

Digital (21, 70, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 70, 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+49, 116)-Net in Base 27 — Constructive

(21, 70, 116)-net in base 27, using

6 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+49, 163)-Net over F₂₇ — Digital

Digital (21, 70, 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+49, 4902)-Net in Base 27 — Upper bound on s

There is no (21, 70, 4903)-net in base 27, because

1 times m-reduction [i] would yield (21, 69, 4903)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 580 899891 663017 876166 336876 695658 406210 822341 475235 000251 426630 111781 992248 000145 282303 041390 487569 > 27⁶⁹ [i]