MinT - Best Known (54−23, 54, s)-Nets in Base 27

Best Known (54−23, 54, s)-Nets in Base 27

(54−23, 54, 196)-Net over F₂₇ — Constructive and digital

Digital (31, 54, 196)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 11, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 4 and N(F) ≥ 64, using
      - a function field by SÃ©mirat [i]
2. digital (4, 15, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (5, 28, 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]

(54−23, 54, 370)-Net in Base 27 — Constructive

(31, 54, 370)-net in base 27, using

6 times m-reduction [i] based on (31, 60, 370)-net in base 27, using
- base change [i] based on digital (16, 45, 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]

(54−23, 54, 1147)-Net over F₂₇ — Digital

Digital (31, 54, 1147)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54−23, 54, 1488024)-Net in Base 27 — Upper bound on s

There is no (31, 54, 1488025)-net in base 27, because

1 times m-reduction [i] would yield (31, 53, 1488025)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 7282 508924 912653 146838 726102 153388 710938 188478 012038 173104 039806 903458 916451 > 27⁵³ [i]