MinT - Best Known (52, 52+39, s)-Nets in Base 27

Best Known (52, 52+39, s)-Nets in Base 27

(52, 52+39, 240)-Net over F₂₇ — Constructive and digital

Digital (52, 91, 240)-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 (7, 26, 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]
3. digital (7, 46, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(52, 52+39, 370)-Net in Base 27 — Constructive

(52, 91, 370)-net in base 27, using

t-expansion [i] based on (43, 91, 370)-net in base 27, using
- 17 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+39, 1566)-Net over F₂₇ — Digital

Digital (52, 91, 1566)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+39, 1838162)-Net in Base 27 — Upper bound on s

There is no (52, 91, 1838163)-net in base 27, because

1 times m-reduction [i] would yield (52, 90, 1838163)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 664 873544 403270 269037 951109 663579 167148 283894 672712 789475 817587 611189 711712 653226 817546 915782 025469 684202 042412 516881 296568 698147 > 27⁹⁰ [i]