MinT - Best Known (56, 56+54, s)-Nets in Base 27

Best Known (56, 56+54, s)-Nets in Base 27

(56, 56+54, 204)-Net over F₂₇ — Constructive and digital

Digital (56, 110, 204)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (11, 38, 96)-net over F₂₇, using
  - net from sequence [i] based on digital (11, 95)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 11 and N(F) ≥ 96, using
      - an explicitly constructive algebraic function field [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]

(56, 56+54, 370)-Net in Base 27 — Constructive

(56, 110, 370)-net in base 27, using

27² times duplication [i] based on (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]

(56, 56+54, 755)-Net over F₂₇ — Digital

Digital (56, 110, 755)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(56, 56+54, 285032)-Net in Base 27 — Upper bound on s

There is no (56, 110, 285033)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 28 186684 487350 520468 783016 196220 194731 312444 386068 675825 555058 483558 240313 270166 120212 222033 562625 655860 934751 181983 644124 785789 714201 154008 339282 200977 828579 > 27¹¹⁰ [i]