MinT - Best Known (99−39, 99, s)-Nets in Base 27

Best Known (99−39, 99, s)-Nets in Base 27

(99−39, 99, 274)-Net over F₂₇ — Constructive and digital

Digital (60, 99, 274)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (4, 13, 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, 17, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. digital (4, 23, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
4. digital (7, 46, 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]

(99−39, 99, 446)-Net in Base 27 — Constructive

(60, 99, 446)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (6, 25, 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]
2. (35, 74, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (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]

(99−39, 99, 3116)-Net over F₂₇ — Digital

Digital (60, 99, 3116)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(99−39, 99, 7363176)-Net in Base 27 — Upper bound on s

There is no (60, 99, 7363177)-net in base 27, because

1 times m-reduction [i] would yield (60, 98, 7363177)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 187 780055 372795 906505 437666 653743 754927 827280 303307 539823 349826 790996 738648 135696 258268 882987 792837 617314 416568 777002 004592 724958 679708 582979 > 27⁹⁸ [i]