MinT - Best Known (109−54, 109, s)-Nets in Base 27

Best Known (109−54, 109, s)-Nets in Base 27

(109−54, 109, 202)-Net over F₂₇ — Constructive and digital

Digital (55, 109, 202)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (10, 37, 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, 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]

(109−54, 109, 370)-Net in Base 27 — Constructive

(55, 109, 370)-net in base 27, using

27¹ 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]

(109−54, 109, 707)-Net over F₂₇ — Digital

Digital (55, 109, 707)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−54, 109, 252277)-Net in Base 27 — Upper bound on s

There is no (55, 109, 252278)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 043982 772147 316851 924944 977309 825478 421727 076415 179710 320105 210652 597407 991391 495630 433547 785985 013644 981644 441586 854487 883944 267459 344942 691541 343895 875649 > 27¹⁰⁹ [i]