MinT - Best Known (108−67, 108, s)-Nets in Base 27

Best Known (108−67, 108, s)-Nets in Base 27

(108−67, 108, 128)-Net over F₂₇ — Constructive and digital

Digital (41, 108, 128)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 37, 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, 71, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)

(108−67, 108, 224)-Net in Base 27 — Constructive

(41, 108, 224)-net in base 27, using

t-expansion [i] based on (40, 108, 224)-net in base 27, using
- base change [i] based on digital (13, 81, 224)-net over F₈₁, using
  - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
      - a function field by SÃ©mirat [i]

(108−67, 108, 273)-Net over F₂₇ — Digital

Digital (41, 108, 273)-net over F₂₇, using

t-expansion [i] based on digital (40, 108, 273)-net over F₂₇, using
- net from sequence [i] based on digital (40, 272)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 40 and N(F) ≥ 273, using
    - a curve from the manYPoints database [i]

(108−67, 108, 298)-Net in Base 27

(41, 108, 298)-net in base 27, using

t-expansion [i] based on (39, 108, 298)-net in base 27, using
- base change [i] based on digital (12, 81, 298)-net over F₈₁, using
  - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
      - a curve from the manYPoints database [i]

(108−67, 108, 22137)-Net in Base 27 — Upper bound on s

There is no (41, 108, 22138)-net in base 27, because

1 times m-reduction [i] would yield (41, 107, 22138)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1432 764826 213154 003705 037285 593571 172425 201828 008418 893764 793063 994863 303357 434302 585055 355898 292375 922663 530873 469949 546632 220770 213561 575892 104629 969189 > 27¹⁰⁷ [i]