MinT - Best Known (45, 45+36, s)-Nets in Base 27

Best Known (45, 45+36, s)-Nets in Base 27

(45, 45+36, 210)-Net over F₂₇ — Constructive and digital

Digital (45, 81, 210)-net over F₂₇, using

1 times m-reduction [i] based on digital (45, 82, 210)-net over F₂₇, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (4, 16, 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, 22, 64)-net over F₂₇, using
    - net from sequence [i] based on digital (4, 63)-sequence over F₂₇ (see above)
  3. digital (7, 44, 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]

(45, 45+36, 370)-Net in Base 27 — Constructive

(45, 81, 370)-net in base 27, using

t-expansion [i] based on (43, 81, 370)-net in base 27, using
- 27 times m-reduction [i] based on (43, 108, 370)-net in base 27, using
  - base change [i] based on digital (16, 81, 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]

(45, 45+36, 1116)-Net over F₂₇ — Digital

Digital (45, 81, 1116)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(45, 45+36, 802211)-Net in Base 27 — Upper bound on s

There is no (45, 81, 802212)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 87 191388 401522 381219 081879 955768 342890 247938 212267 982217 267245 416434 774508 385564 675047 061545 437806 369617 751701 119225 > 27⁸¹ [i]