MinT - Best Known (87−47, 87, s)-Nets in Base 27

Best Known (87−47, 87, s)-Nets in Base 27

(87−47, 87, 176)-Net over F₂₇ — Constructive and digital

Digital (40, 87, 176)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 30, 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 (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]

(87−47, 87, 350)-Net over F₂₇ — Digital

Digital (40, 87, 350)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−47, 87, 370)-Net in Base 27 — Constructive

(40, 87, 370)-net in base 27, using

9 times m-reduction [i] based on (40, 96, 370)-net in base 27, using
- base change [i] based on digital (16, 72, 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]

(87−47, 87, 81559)-Net in Base 27 — Upper bound on s

There is no (40, 87, 81560)-net in base 27, because

1 times m-reduction [i] would yield (40, 86, 81560)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1251 191946 743704 287970 755971 023758 067783 101544 594313 773987 458830 885504 982925 489764 954707 052897 595556 977231 565819 658216 360033 > 27⁸⁶ [i]