MinT - Best Known (56, 56+41, s)-Nets in Base 27

Best Known (56, 56+41, s)-Nets in Base 27

(56, 56+41, 252)-Net over F₂₇ — Constructive and digital

Digital (56, 97, 252)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 20, 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, 27, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (9, 50, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(56, 56+41, 408)-Net in Base 27 — Constructive

(56, 97, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 21, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (35, 76, 370)-net in base 27, using
  - base change [i] based on digital (16, 57, 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]

(56, 56+41, 1814)-Net over F₂₇ — Digital

Digital (56, 97, 1814)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+41, 2370702)-Net in Base 27 — Upper bound on s

There is no (56, 97, 2370703)-net in base 27, because

1 times m-reduction [i] would yield (56, 96, 2370703)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 257586 413875 933040 559636 614490 989300 359110 350147 795038 204080 182151 829962 342048 457532 084737 990087 759910 357475 452414 492041 015381 369796 743161 > 27⁹⁶ [i]