MinT - Best Known (87−35, 87, s)-Nets in Base 27

Best Known (87−35, 87, s)-Nets in Base 27

(87−35, 87, 258)-Net over F₂₇ — Constructive and digital

Digital (52, 87, 258)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 18, 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, 24, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (10, 45, 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]

(87−35, 87, 418)-Net in Base 27 — Constructive

(52, 87, 418)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (2, 19, 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. (33, 68, 370)-net in base 27, using
  - base change [i] based on digital (16, 51, 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]

(87−35, 87, 2411)-Net over F₂₇ — Digital

Digital (52, 87, 2411)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(87−35, 87, 4808258)-Net in Base 27 — Upper bound on s

There is no (52, 87, 4808259)-net in base 27, because

1 times m-reduction [i] would yield (52, 86, 4808259)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1251 080319 164696 099241 228671 355728 232808 820223 152257 228388 712108 019733 796049 159900 463782 479215 446407 012525 574312 329135 020975 > 27⁸⁶ [i]