MinT - Best Known (101−55, 101, s)-Nets in Base 27

Best Known (101−55, 101, s)-Nets in Base 27

(101−55, 101, 182)-Net over F₂₇ — Constructive and digital

Digital (46, 101, 182)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 36, 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, 65, 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]

(101−55, 101, 370)-Net in Base 27 — Constructive

(46, 101, 370)-net in base 27, using

t-expansion [i] based on (43, 101, 370)-net in base 27, using
- 7 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]

(101−55, 101, 376)-Net over F₂₇ — Digital

Digital (46, 101, 376)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(101−55, 101, 84082)-Net in Base 27 — Upper bound on s

There is no (46, 101, 84083)-net in base 27, because

1 times m-reduction [i] would yield (46, 100, 84083)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 136891 501350 776672 533239 623085 513832 929136 576587 893081 651408 784627 102669 906267 506893 868815 000161 569926 052146 314286 602325 938092 642393 860662 694995 > 27¹⁰⁰ [i]