MinT - Best Known (54, 106, s)-Nets in Base 27

Best Known (54, 106, s)-Nets in Base 27

(54, 106, 202)-Net over F₂₇ — Constructive and digital

Digital (54, 106, 202)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 36, 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]
2. digital (18, 70, 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, 106, 370)-Net in Base 27 — Constructive

(54, 106, 370)-net in base 27, using

t-expansion [i] based on (43, 106, 370)-net in base 27, using
- 2 times m-reduction [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, 106, 737)-Net over F₂₇ — Digital

Digital (54, 106, 737)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(54, 106, 277880)-Net in Base 27 — Upper bound on s

There is no (54, 106, 277881)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 53 035510 303568 200405 067821 530210 938449 088404 822748 719433 702831 291033 854139 341449 847979 109210 777910 051999 345931 874448 891961 068773 367547 660188 552583 458089 > 27¹⁰⁶ [i]