MinT - Best Known (57, 57+44, s)-Nets in Base 27

Best Known (57, 57+44, s)-Nets in Base 27

(57, 57+44, 246)-Net over F₂₇ — Constructive and digital

Digital (57, 101, 246)-net over F₂₇, using

generalized (u,Â u+v)-construction [i] based on
1. digital (7, 21, 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 (7, 51, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(57, 57+44, 370)-Net in Base 27 — Constructive

(57, 101, 370)-net in base 27, using

t-expansion [i] based on (43, 101, 370)-net in base 27, using
- 7 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]

(57, 57+44, 1516)-Net over F₂₇ — Digital

Digital (57, 101, 1516)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(57, 57+44, 1297587)-Net in Base 27 — Upper bound on s

There is no (57, 101, 1297588)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 696074 014349 502134 837007 108720 708657 360324 512026 642021 844514 702374 958968 180700 635849 179215 507719 480493 174851 104072 882698 756160 216848 306454 614777 > 27¹⁰¹ [i]