MinT - Best Known (44, 109, s)-Nets in Base 27

Best Known (44, 109, s)-Nets in Base 27

(44, 109, 152)-Net over F₂₇ — Constructive and digital

Digital (44, 109, 152)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (6, 38, 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, 71, 76)-net over F₂₇, using
  - net from sequence [i] based on digital (6, 75)-sequence over F₂₇ (see above)

(44, 109, 280)-Net over F₂₇ — Digital

Digital (44, 109, 280)-net over F₂₇, using

t-expansion [i] based on digital (42, 109, 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]

(44, 109, 370)-Net in Base 27 — Constructive

(44, 109, 370)-net in base 27, using

27¹ times duplication [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]

(44, 109, 33307)-Net in Base 27 — Upper bound on s

There is no (44, 109, 33308)-net in base 27, because

1 times m-reduction [i] would yield (44, 108, 33308)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38681 863542 625590 400974 101654 057888 129601 490746 703662 351233 824264 905002 248125 404211 511476 507292 407490 328248 309932 114130 685593 808556 290564 930352 304619 874561 > 27¹⁰⁸ [i]