MinT - Best Known (38−15, 38, s)-Nets in Base 27

Best Known (38−15, 38, s)-Nets in Base 27

(38−15, 38, 224)-Net over F₂₇ — Constructive and digital

Digital (23, 38, 224)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (0, 1, 28)-net over F₂₇, using
  - s-reduction based on digital (0, 1, s)-net over F₂₇ with arbitrarily large s, using
    - construction for strength kÂ =Â 1 [i]
2. digital (0, 2, 28)-net over F₂₇, using
  - kÂ =Â 2 construction [i]
3. digital (0, 2, 28)-net over F₂₇ (see above)
4. digital (0, 3, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 0 and N(F) ≥ 28, using
      - the rational function field F₂₇(x) [i]
    - Niederreiter sequence [i]
5. digital (0, 3, 28)-net over F₂₇ (see above)
6. digital (0, 5, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
7. digital (0, 7, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
8. digital (0, 15, 28)-net over F₂₇, using
  - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)

(38−15, 38, 266)-Net in Base 27 — Constructive

(23, 38, 266)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. (5, 12, 116)-net in base 27, using
  - base change [i] based on digital (2, 9, 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]
2. (11, 26, 150)-net in base 27, using
  - 2 times m-reduction [i] based on (11, 28, 150)-net in base 27, using
    - base change [i] based on digital (4, 21, 150)-net over F₈₁, using
      - net from sequence [i] based on digital (4, 149)-sequence over F₈₁, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 4 and N(F) ≥ 150, using
        a function field by SÃ©mirat [i]

(38−15, 38, 1792)-Net over F₂₇ — Digital

Digital (23, 38, 1792)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(38−15, 38, 4783270)-Net in Base 27 — Upper bound on s

There is no (23, 38, 4783271)-net in base 27, because

1 times m-reduction [i] would yield (23, 37, 4783271)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 91297 710178 490258 474828 581765 072096 188224 393901 639339 > 27³⁷ [i]