MinT - Best Known (90−47, 90, s)-Nets in Base 27

Best Known (90−47, 90, s)-Nets in Base 27

(90−47, 90, 188)-Net over F₂₇ — Constructive and digital

Digital (43, 90, 188)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 33, 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, 57, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(90−47, 90, 370)-Net in Base 27 — Constructive

(43, 90, 370)-net in base 27, using

18 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]

(90−47, 90, 440)-Net over F₂₇ — Digital

Digital (43, 90, 440)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(90−47, 90, 125371)-Net in Base 27 — Upper bound on s

There is no (43, 90, 125372)-net in base 27, because

1 times m-reduction [i] would yield (43, 89, 125372)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 24 629303 801036 784158 529946 195252 209391 280207 046412 485878 766113 254332 911506 351960 347897 547963 598929 908101 393099 334157 717599 552945 > 27⁸⁹ [i]