MinT - Best Known (109−49, 109, s)-Nets in Base 27

Best Known (109−49, 109, s)-Nets in Base 27

(109−49, 109, 240)-Net over F₂₇ — Constructive and digital

Digital (60, 109, 240)-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 (7, 31, 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]
3. digital (7, 56, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(109−49, 109, 370)-Net in Base 27 — Constructive

(60, 109, 370)-net in base 27, using

27¹ times duplication [i] based on (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]

(109−49, 109, 1307)-Net over F₂₇ — Digital

Digital (60, 109, 1307)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−49, 109, 1041168)-Net in Base 27 — Upper bound on s

There is no (60, 109, 1041169)-net in base 27, because

1 times m-reduction [i] would yield (60, 108, 1041169)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38662 375894 601638 347519 438883 686328 265210 310868 028972 336112 799340 757148 072297 137857 207824 101465 267743 295865 762801 643029 550705 413992 865150 882071 364864 680609 > 27¹⁰⁸ [i]