MinT - Best Known (42, 42+44, s)-Nets in Base 27

Best Known (42, 42+44, s)-Nets in Base 27

(42, 42+44, 188)-Net over F₂₇ — Constructive and digital

Digital (42, 86, 188)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 32, 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]
2. digital (10, 54, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(42, 42+44, 370)-Net in Base 27 — Constructive

(42, 86, 370)-net in base 27, using

18 times m-reduction [i] based on (42, 104, 370)-net in base 27, using
- base change [i] based on digital (16, 78, 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]

(42, 42+44, 482)-Net over F₂₇ — Digital

Digital (42, 86, 482)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(42, 42+44, 137143)-Net in Base 27 — Upper bound on s

There is no (42, 86, 137144)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1251 201741 774921 617473 547042 084912 032582 979885 503059 465654 996499 443520 970772 152067 036652 640203 846371 907650 101980 217717 006673 > 27⁸⁶ [i]