MinT - Best Known (21, 21+43, s)-Nets in Base 27

Best Known (21, 21+43, s)-Nets in Base 27

(21, 21+43, 108)-Net over F₂₇ — Constructive and digital

Digital (21, 64, 108)-net over F₂₇, using

t-expansion [i] based on digital (18, 64, 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+43, 160)-Net in Base 27 — Constructive

(21, 64, 160)-net in base 27, using

base change [i] based on digital (5, 48, 160)-net over F₈₁, using
- net from sequence [i] based on digital (5, 159)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 160, using
    - a function field by SÃ©mirat [i]

(21, 21+43, 163)-Net over F₂₇ — Digital

Digital (21, 64, 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+43, 167)-Net in Base 27

(21, 64, 167)-net in base 27, using

base change [i] based on digital (5, 48, 167)-net over F₈₁, using
- net from sequence [i] based on digital (5, 166)-sequence over F₈₁, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 5 and N(F) ≥ 167, using
    - a curve from the manYPoints database [i]

(21, 21+43, 6559)-Net in Base 27 — Upper bound on s

There is no (21, 64, 6560)-net in base 27, because

1 times m-reduction [i] would yield (21, 63, 6560)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 499977 165983 449373 477530 724502 927506 350584 916541 419963 649764 844052 626920 513588 546399 272513 > 27⁶³ [i]