MinT - Best Known (74−29, 74, s)-Nets in Base 27

Best Known (74−29, 74, s)-Nets in Base 27

(74−29, 74, 248)-Net over F₂₇ — Constructive and digital

Digital (45, 74, 248)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 10, 56)-net over F₂₇, using
  - (u,Â u+v)-construction [i] based on
    1. 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]
    2. digital (0, 7, 28)-net over F₂₇, using
      - net from sequence [i] based on digital (0, 27)-sequence over F₂₇ (see above)
2. digital (4, 13, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
3. digital (4, 18, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
4. digital (4, 33, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(74−29, 74, 398)-Net in Base 27 — Constructive

(45, 74, 398)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (0, 14, 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]
2. (31, 60, 370)-net in base 27, using
  - base change [i] based on digital (16, 45, 370)-net over F₈₁, using
    - net from sequence [i] based on digital (16, 369)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 16 and N(F) ≥ 370, using
        a function field by GarcÃa and Quoos [i]

(74−29, 74, 2650)-Net over F₂₇ — Digital

Digital (45, 74, 2650)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(74−29, 74, 6761196)-Net in Base 27 — Upper bound on s

There is no (45, 74, 6761197)-net in base 27, because

1 times m-reduction [i] would yield (45, 73, 6761197)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 308 713136 595197 294751 991337 208392 401619 721817 431201 658316 025094 423850 971371 506660 470961 178550 542093 711393 > 27⁷³ [i]