MinT - Best Known (57, 98, s)-Nets in Base 27

Best Known (57, 98, s)-Nets in Base 27

(57, 98, 258)-Net over F₂₇ — Constructive and digital

Digital (57, 98, 258)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 20, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 7 and N(F) ≥ 82, using
      - a function field by SÃ©mirat [i]
2. digital (7, 27, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (10, 51, 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]

(57, 98, 418)-Net in Base 27 — Constructive

(57, 98, 418)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (2, 22, 48)-net over F₂₇, using
  - net from sequence [i] based on digital (2, 47)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 2 and N(F) ≥ 48, using
      - a function field by GarcÃa and Quoos [i]
2. (35, 76, 370)-net in base 27, using
  - base change [i] based on digital (16, 57, 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]

(57, 98, 1968)-Net over F₂₇ — Digital

Digital (57, 98, 1968)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 98, 2795410)-Net in Base 27 — Upper bound on s

There is no (57, 98, 2795411)-net in base 27, because

1 times m-reduction [i] would yield (57, 97, 2795411)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 6 954848 064231 652537 556476 131691 938059 843757 754161 173480 405326 426249 830385 603575 943019 741984 191461 498572 095763 285909 021024 173778 966986 564569 > 27⁹⁷ [i]