MinT - Best Known (106−63, 106, s)-Nets in Base 27

Best Known (106−63, 106, s)-Nets in Base 27

(106−63, 106, 152)-Net over F₂₇ — Constructive and digital

Digital (43, 106, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 37, 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]
2. digital (6, 69, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(106−63, 106, 280)-Net over F₂₇ — Digital

Digital (43, 106, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 106, 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]

(106−63, 106, 370)-Net in Base 27 — Constructive

(43, 106, 370)-net in base 27, using

2 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]

(106−63, 106, 33652)-Net in Base 27 — Upper bound on s

There is no (43, 106, 33653)-net in base 27, because

1 times m-reduction [i] would yield (43, 105, 33653)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 1 965010 914003 543137 948317 619549 219234 520481 061097 416397 978394 138377 899977 592733 510048 634828 403370 809526 686959 829701 166141 032915 318706 724682 001013 570819 > 27¹⁰⁵ [i]