MinT - Best Known (69−30, 69, s)-Nets in Base 27

Best Known (69−30, 69, s)-Nets in Base 27

(69−30, 69, 204)-Net over F₂₇ — Constructive and digital

Digital (39, 69, 204)-net over F₂₇, using

1 times m-reduction [i] based on digital (39, 70, 204)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 14, 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, 19, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (6, 37, 76)-net over F₂₇, using
    - net from sequence [i] based on digital (6, 75)-sequence over F₂₇, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 6 and N(F) ≥ 76, using
        a function field by SÃ©mirat [i]

(69−30, 69, 370)-Net in Base 27 — Constructive

(39, 69, 370)-net in base 27, using

23 times m-reduction [i] based on (39, 92, 370)-net in base 27, using
- base change [i] based on digital (16, 69, 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]

(69−30, 69, 1157)-Net over F₂₇ — Digital

Digital (39, 69, 1157)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(69−30, 69, 948556)-Net in Base 27 — Upper bound on s

There is no (39, 69, 948557)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 580 903701 825085 273986 479623 868734 321724 362912 918542 925308 107730 559523 659020 843269 248130 659263 361635 > 27⁶⁹ [i]