MinT - Best Known (109−46, 109, s)-Nets in Base 27

Best Known (109−46, 109, s)-Nets in Base 27

(109−46, 109, 260)-Net over F₂₇ — Constructive and digital

Digital (63, 109, 260)-net over F₂₇, using

1 times m-reduction [i] based on digital (63, 110, 260)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (7, 22, 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 (8, 31, 84)-net over F₂₇, using
    - net from sequence [i] based on digital (8, 83)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 8 and N(F) ≥ 84, using
        a function field by SÃ©mirat [i]
  3. digital (10, 57, 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]

(109−46, 109, 422)-Net in Base 27 — Constructive

(63, 109, 422)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (3, 26, 52)-net over F₂₇, using
  - net from sequence [i] based on digital (3, 51)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 3 and N(F) ≥ 52, using
      - a function field by SÃ©mirat [i]
2. (37, 83, 370)-net in base 27, using
  - 1 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
    - base change [i] based on digital (16, 63, 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]

(109−46, 109, 2010)-Net over F₂₇ — Digital

Digital (63, 109, 2010)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−46, 109, 2202420)-Net in Base 27 — Upper bound on s

There is no (63, 109, 2202421)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 043888 904675 559842 577858 758550 933708 449012 894759 104883 702361 180912 489368 264732 817010 202046 706377 844436 076768 865720 361537 014699 998604 572636 708762 196702 305443 > 27¹⁰⁹ [i]