MinT - Best Known (54, 54+43, s)-Nets in Base 27

Best Known (54, 54+43, s)-Nets in Base 27

(54, 54+43, 234)-Net over F₂₇ — Constructive and digital

Digital (54, 97, 234)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 20, 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, 27, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (7, 50, 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]

(54, 54+43, 370)-Net in Base 27 — Constructive

(54, 97, 370)-net in base 27, using

t-expansion [i] based on (43, 97, 370)-net in base 27, using
- 11 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]

(54, 54+43, 1305)-Net over F₂₇ — Digital

Digital (54, 97, 1305)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+43, 1166520)-Net in Base 27 — Upper bound on s

There is no (54, 97, 1166521)-net in base 27, because

1 times m-reduction [i] would yield (54, 96, 1166521)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 257587 440479 574911 759657 374217 981146 179383 807841 819488 816178 275083 991107 180675 768763 901902 246880 299488 946595 629309 477623 840505 613926 758107 > 27⁹⁶ [i]