MinT - Best Known (32, 32+37, s)-Nets in Base 27

Best Known (32, 32+37, s)-Nets in Base 27

(32, 32+37, 164)-Net over F₂₇ — Constructive and digital

Digital (32, 69, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 25, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (7, 44, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(32, 32+37, 224)-Net in Base 27 — Constructive

(32, 69, 224)-net in base 27, using

7 times m-reduction [i] based on (32, 76, 224)-net in base 27, using
- base change [i] based on digital (13, 57, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(32, 32+37, 309)-Net over F₂₇ — Digital

Digital (32, 69, 309)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(32, 32+37, 74212)-Net in Base 27 — Upper bound on s

There is no (32, 69, 74213)-net in base 27, because

1 times m-reduction [i] would yield (32, 68, 74213)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 21 517272 475321 181721 734235 062368 934522 899981 731405 585844 081942 805615 006499 303917 067413 427389 584929 > 27⁶⁸ [i]