MinT - Best Known (54, 54+38, s)-Nets in Base 27

Best Known (54, 54+38, s)-Nets in Base 27

(54, 54+38, 252)-Net over F₂₇ — Constructive and digital

Digital (54, 92, 252)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 19, 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 (7, 26, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (9, 47, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(54, 54+38, 408)-Net in Base 27 — Constructive

(54, 92, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 20, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (34, 72, 370)-net in base 27, using
  - base change [i] based on digital (16, 54, 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]

(54, 54+38, 2060)-Net over F₂₇ — Digital

Digital (54, 92, 2060)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+38, 2600486)-Net in Base 27 — Upper bound on s

There is no (54, 92, 2600487)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 484695 425335 751934 186607 955738 346374 929789 912246 543246 480149 543552 803640 221799 420055 508682 367758 948510 606746 230169 059057 758061 806579 > 27⁹² [i]