MinT - Best Known (39, 64, s)-Nets in Base 27

Best Known (39, 64, s)-Nets in Base 27

(39, 64, 234)-Net over F₂₇ — Constructive and digital

Digital (39, 64, 234)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 14, 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, 18, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (7, 32, 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]

(39, 64, 370)-Net in Base 27 — Constructive

(39, 64, 370)-net in base 27, using

28 times m-reduction [i] based on (39, 92, 370)-net in base 27, using
- base change [i] based on digital (16, 69, 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]

(39, 64, 2486)-Net over F₂₇ — Digital

Digital (39, 64, 2486)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(39, 64, 6653458)-Net in Base 27 — Upper bound on s

There is no (39, 64, 6653459)-net in base 27, because

1 times m-reduction [i] would yield (39, 63, 6653459)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 499400 672373 770003 949894 610283 344695 729340 018146 881922 244397 879751 098542 739077 104171 611177 > 27⁶³ [i]