MinT - Best Known (82−36, 82, s)-Nets in Base 27

Best Known (82−36, 82, s)-Nets in Base 27

(82−36, 82, 216)-Net over F₂₇ — Constructive and digital

Digital (46, 82, 216)-net over F₂₇, using

1 times m-reduction [i] based on digital (46, 83, 216)-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 (6, 24, 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]
  3. digital (6, 43, 76)-net over F₂₇, using
    - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(82−36, 82, 370)-Net in Base 27 — Constructive

(46, 82, 370)-net in base 27, using

t-expansion [i] based on (43, 82, 370)-net in base 27, using
- 26 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]

(82−36, 82, 1225)-Net over F₂₇ — Digital

Digital (46, 82, 1225)-net over F₂₇, using

Gilbertâ€“VarÅ¡amov bound [i]

(82−36, 82, 963406)-Net in Base 27 — Upper bound on s

There is no (46, 82, 963407)-net in base 27, because

the generalized Rao bound for nets shows that 27^mÂ â‰¥ 2354 128616 169132 866222 880256 187798 970715 682739 197733 390861 077679 890726 224581 626972 259033 534479 631592 704424 314252 749005 > 27⁸² [i]