MinT - Best Known (44, 44+51, s)-Nets in Base 27

Best Known (44, 44+51, s)-Nets in Base 27

(44, 44+51, 182)-Net over F₂₇ — Constructive and digital

Digital (44, 95, 182)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 34, 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]
2. digital (10, 61, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(44, 44+51, 370)-Net in Base 27 — Constructive

(44, 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]

(44, 44+51, 389)-Net over F₂₇ — Digital

Digital (44, 95, 389)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(44, 44+51, 94301)-Net in Base 27 — Upper bound on s

There is no (44, 95, 94302)-net in base 27, because

1 times m-reduction [i] would yield (44, 94, 94302)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 353 348110 637517 349135 280219 743355 344926 036344 681391 536791 703785 927912 036027 763281 588266 929314 761971 152104 487272 986732 709963 389826 116829 > 27⁹⁴ [i]