MinT - Best Known (44, 44+31, s)-Nets in Base 27

Best Known (44, 44+31, s)-Nets in Base 27

(44, 44+31, 234)-Net over F₂₇ — Constructive and digital

Digital (44, 75, 234)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 16, 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, 21, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (7, 38, 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, 44+31, 370)-Net in Base 27 — Constructive

(44, 75, 370)-net in base 27, using

t-expansion [i] based on (43, 75, 370)-net in base 27, using
- 33 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+31, 1770)-Net over F₂₇ — Digital

Digital (44, 75, 1770)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(44, 44+31, 2845684)-Net in Base 27 — Upper bound on s

There is no (44, 75, 2845685)-net in base 27, because

1 times m-reduction [i] would yield (44, 74, 2845685)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 8335 282499 225972 284819 593108 657925 437780 929037 302443 812466 452284 681686 473612 958831 728807 833517 963752 461635 > 27⁷⁴ [i]