MinT - Best Known (102−59, 102, s)-Nets in Base 27

Best Known (102−59, 102, s)-Nets in Base 27

(102−59, 102, 164)-Net over F₂₇ — Constructive and digital

Digital (43, 102, 164)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (7, 36, 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, 66, 82)-net over F₂₇, using
  - net from sequence [i] based on digital (7, 81)-sequence over F₂₇ (see above)

(102−59, 102, 280)-Net over F₂₇ — Digital

Digital (43, 102, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 102, 280)-net over F₂₇, using
- net from sequence [i] based on digital (42, 279)-sequence over F₂₇, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 42 and N(F) ≥ 280, using
    - Drinfeld modules of rank 1 [i]

(102−59, 102, 370)-Net in Base 27 — Constructive

(43, 102, 370)-net in base 27, using

6 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]

(102−59, 102, 43360)-Net in Base 27 — Upper bound on s

There is no (43, 102, 43361)-net in base 27, because

1 times m-reduction [i] would yield (43, 101, 43361)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 3 697915 114439 049942 891436 014755 928882 169752 261548 814287 001709 272658 333364 158147 746988 755332 623668 779848 082496 035005 509733 050543 938098 690709 474571 > 27¹⁰¹ [i]