MinT - Best Known (52, 52+40, s)-Nets in Base 27

Best Known (52, 52+40, s)-Nets in Base 27

(52, 52+40, 234)-Net over F₂₇ — Constructive and digital

Digital (52, 92, 234)-net over F₂₇, using

1 times m-reduction [i] based on digital (52, 93, 234)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (6, 19, 76)-net over F₂₇, using
    - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]
  2. digital (6, 26, 76)-net over F₂₇, using
    - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
  3. digital (7, 48, 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]

(52, 52+40, 370)-Net in Base 27 — Constructive

(52, 92, 370)-net in base 27, using

t-expansion [i] based on (43, 92, 370)-net in base 27, using
- 16 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 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]

(52, 52+40, 1429)-Net over F₂₇ — Digital

Digital (52, 92, 1429)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+40, 1226316)-Net in Base 27 — Upper bound on s

There is no (52, 92, 1226317)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484696 689651 101477 345403 986695 466562 041891 989764 486327 778313 534083 858172 976371 675084 510437 167441 952028 977035 078457 846973 095520 810097 > 27⁹² [i]