MinT - Best Known (98−38, 98, s)-Nets in Base 27

Best Known (98−38, 98, s)-Nets in Base 27

(98−38, 98, 280)-Net over F₂₇ — Constructive and digital

Digital (60, 98, 280)-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, 16, 64)-net over F₂₇, using
  - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
3. 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]
4. digital (6, 44, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(98−38, 98, 452)-Net in Base 27 — Constructive

(60, 98, 452)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (7, 26, 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. (34, 72, 370)-net in base 27, using
  - base change [i] based on digital (16, 54, 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]

(98−38, 98, 3503)-Net over F₂₇ — Digital

Digital (60, 98, 3503)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(98−38, 98, 7363176)-Net in Base 27 — Upper bound on s

There is no (60, 98, 7363177)-net in base 27, because

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]