MinT - Best Known (49, 49+42, s)-Nets in Base 27

Best Known (49, 49+42, s)-Nets in Base 27

(49, 49+42, 204)-Net over F₂₇ — Constructive and digital

Digital (49, 91, 204)-net over F₂₇, using

1 times m-reduction [i] based on digital (49, 92, 204)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 18, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
        a function field by SÃ©mirat [i]
  2. digital (4, 25, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (6, 49, 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]

(49, 49+42, 370)-Net in Base 27 — Constructive

(49, 91, 370)-net in base 27, using

t-expansion [i] based on (43, 91, 370)-net in base 27, using
- 17 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]

(49, 49+42, 954)-Net over F₂₇ — Digital

Digital (49, 91, 954)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(49, 49+42, 532214)-Net in Base 27 — Upper bound on s

There is no (49, 91, 532215)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 17951 920609 829267 897211 581481 277883 422796 202025 957987 224539 621298 561947 337749 912966 683481 990526 192047 949226 660959 675907 153023 597519 > 27⁹¹ [i]