MinT - Best Known (101−41, 101, s)-Nets in Base 27

Best Known (101−41, 101, s)-Nets in Base 27

(101−41, 101, 270)-Net over F₂₇ — Constructive and digital

Digital (60, 101, 270)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 20, 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, 30, 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]
3. digital (10, 51, 94)-net over F₂₇, using
  - net from sequence [i] based on digital (10, 93)-sequence over F₂₇ (see above)

(101−41, 101, 438)-Net in Base 27 — Constructive

(60, 101, 438)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (5, 25, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]
2. (35, 76, 370)-net in base 27, using
  - base change [i] based on digital (16, 57, 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]

(101−41, 101, 2515)-Net over F₂₇ — Digital

Digital (60, 101, 2515)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(101−41, 101, 4583009)-Net in Base 27 — Upper bound on s

There is no (60, 101, 4583010)-net in base 27, because

1 times m-reduction [i] would yield (60, 100, 4583010)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 136891 947616 373912 495454 969409 620868 095278 245932 917638 680617 679449 001187 599855 920462 194532 208561 663964 310843 190232 913881 605468 210892 275400 400073 > 27¹⁰⁰ [i]