MinT - Best Known (44, 77, s)-Nets in Base 27

Best Known (44, 77, s)-Nets in Base 27

(44, 77, 222)-Net over F₂₇ — Constructive and digital

Digital (44, 77, 222)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 15, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (6, 22, 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]
3. digital (7, 40, 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]

(44, 77, 370)-Net in Base 27 — Constructive

(44, 77, 370)-net in base 27, using

t-expansion [i] based on (43, 77, 370)-net in base 27, using
- 31 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, 77, 1384)-Net over F₂₇ — Digital

Digital (44, 77, 1384)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(44, 77, 1646421)-Net in Base 27 — Upper bound on s

There is no (44, 77, 1646422)-net in base 27, because

1 times m-reduction [i] would yield (44, 76, 1646422)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 076406 151112 442370 259037 699546 022585 208397 504836 570483 017340 374182 611583 841687 392133 915438 677132 959112 307873 > 27⁷⁶ [i]