MinT - Best Known (60, 105, s)-Nets in Base 27

Best Known (60, 105, s)-Nets in Base 27

(60, 105, 252)-Net over F₂₇ — Constructive and digital

Digital (60, 105, 252)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 22, 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, 29, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)
3. digital (9, 54, 88)-net over F₂₇, using
  - net from sequence [i] based on digital (9, 87)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 9 and N(F) ≥ 88, using
      - a function field by SÃ©mirat [i]

(60, 105, 408)-Net in Base 27 — Constructive

(60, 105, 408)-net in base 27, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 38)-net over F₂₇, using
  - net from sequence [i] based on digital (1, 37)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 1 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. (37, 82, 370)-net in base 27, using
  - 2 times m-reduction [i] based on (37, 84, 370)-net in base 27, using
    - base change [i] based on digital (16, 63, 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]

(60, 105, 1751)-Net over F₂₇ — Digital

Digital (60, 105, 1751)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(60, 105, 2033874)-Net in Base 27 — Upper bound on s

There is no (60, 105, 2033875)-net in base 27, because

1 times m-reduction [i] would yield (60, 104, 2033875)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 72750 174792 899589 926934 094532 324105 528467 850723 049442 330690 362350 881586 332348 778955 891467 006605 822637 688771 730239 614207 835249 079370 693943 409405 513101 > 27¹⁰⁴ [i]