MinT - Best Known (59, 108, s)-Nets in Base 27

Best Known (59, 108, s)-Nets in Base 27

(59, 108, 234)-Net over F₂₇ — Constructive and digital

Digital (59, 108, 234)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. 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]
2. digital (6, 30, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)
3. digital (7, 56, 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]

(59, 108, 370)-Net in Base 27 — Constructive

(59, 108, 370)-net in base 27, using

t-expansion [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]

(59, 108, 1222)-Net over F₂₇ — Digital

Digital (59, 108, 1222)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 108, 907570)-Net in Base 27 — Upper bound on s

There is no (59, 108, 907571)-net in base 27, because

1 times m-reduction [i] would yield (59, 107, 907571)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1431 963668 327497 690592 587447 396107 125791 243409 311616 985903 197999 054532 507804 717000 968872 493792 210538 344903 760212 527910 604037 530694 754093 081233 592349 722001 > 27¹⁰⁷ [i]