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

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

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

Digital (54, 108, 196)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (9, 36, 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]
2. digital (18, 72, 108)-net over F₂₇, using
  - net from sequence [i] based on digital (18, 107)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 18 and N(F) ≥ 108, using
      - F₃ from the tower of function fields by Bezerra, GarcÃa, and Stichtenoth over F₂₇ [i]

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

(54, 108, 370)-net in base 27, using

t-expansion [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]

(54, 54+54, 663)-Net over F₂₇ — Digital

Digital (54, 108, 663)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 54+54, 223285)-Net in Base 27 — Upper bound on s

There is no (54, 108, 223286)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38662 963231 402359 766608 245672 826642 440414 520343 426301 633343 034274 258012 898709 723808 674631 908005 220867 277430 982566 657418 161717 053226 871458 545679 599079 128897 > 27¹⁰⁸ [i]