MinT - Best Known (101−57, 101, s)-Nets in Base 27

Best Known (101−57, 101, s)-Nets in Base 27

(101−57, 101, 170)-Net over F₂₇ — Constructive and digital

Digital (44, 101, 170)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 35, 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]
2. digital (9, 66, 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]

(101−57, 101, 308)-Net over F₂₇ — Digital

Digital (44, 101, 308)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(101−57, 101, 370)-Net in Base 27 — Constructive

(44, 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]

(101−57, 101, 56225)-Net in Base 27 — Upper bound on s

There is no (44, 101, 56226)-net in base 27, because

1 times m-reduction [i] would yield (44, 100, 56226)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 136932 398640 884303 008581 594636 893769 743495 329486 833338 696917 550477 859418 297217 743738 076994 169155 655921 472137 459085 780036 525950 301856 510773 896473 > 27¹⁰⁰ [i]