MinT - Best Known (104−57, 104, s)-Nets in Base 27

Best Known (104−57, 104, s)-Nets in Base 27

(104−57, 104, 182)-Net over F₂₇ — Constructive and digital

Digital (47, 104, 182)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 37, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]
2. digital (10, 67, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 10 and N(F) ≥ 94, using
      - a function field by SÃ©mirat [i]

(104−57, 104, 370)-Net in Base 27 — Constructive

(47, 104, 370)-net in base 27, using

t-expansion [i] based on (43, 104, 370)-net in base 27, using
- 4 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]

(104−57, 104, 371)-Net over F₂₇ — Digital

Digital (47, 104, 371)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(104−57, 104, 80043)-Net in Base 27 — Upper bound on s

There is no (47, 104, 80044)-net in base 27, because

1 times m-reduction [i] would yield (47, 103, 80044)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2694 840589 255275 973971 959874 184322 408487 632182 954607 584853 792166 113211 739399 781438 761015 208786 580018 798762 574575 875342 394035 897226 402596 719450 676289 > 27¹⁰³ [i]