MinT - Best Known (90−39, 90, s)-Nets in Base 27

Best Known (90−39, 90, s)-Nets in Base 27

(90−39, 90, 234)-Net over F₂₇ — Constructive and digital

Digital (51, 90, 234)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 19, 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, 25, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (7, 46, 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]

(90−39, 90, 370)-Net in Base 27 — Constructive

(51, 90, 370)-net in base 27, using

t-expansion [i] based on (43, 90, 370)-net in base 27, using
- 18 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]

(90−39, 90, 1438)-Net over F₂₇ — Digital

Digital (51, 90, 1438)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−39, 90, 1545427)-Net in Base 27 — Upper bound on s

There is no (51, 90, 1545428)-net in base 27, because

1 times m-reduction [i] would yield (51, 89, 1545428)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 625082 816558 709770 528628 725695 375621 497693 894215 035281 308594 379213 641834 510001 378024 253370 718087 778696 168107 875701 094589 776337 > 27⁸⁹ [i]