MinT - Best Known (54, 54+41, s)-Nets in Base 27

Best Known (54, 54+41, s)-Nets in Base 27

(54, 54+41, 246)-Net over F₂₇ — Constructive and digital

Digital (54, 95, 246)-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 (7, 48, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(54, 54+41, 370)-Net in Base 27 — Constructive

(54, 95, 370)-net in base 27, using

t-expansion [i] based on (43, 95, 370)-net in base 27, using
- 13 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+41, 1542)-Net over F₂₇ — Digital

Digital (54, 95, 1542)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+41, 1705061)-Net in Base 27 — Upper bound on s

There is no (54, 95, 1705062)-net in base 27, because

1 times m-reduction [i] would yield (54, 94, 1705062)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 353 344362 734492 518225 616602 431534 639795 926375 778474 704259 262268 257282 337599 078900 845298 534853 808711 546456 972616 182149 651822 446704 684969 > 27⁹⁴ [i]