MinT - Best Known (42, 109, s)-Nets in Base 27

Best Known (42, 109, s)-Nets in Base 27

(42, 109, 132)-Net over F₂₇ — Constructive and digital

Digital (42, 109, 132)-net over F₂₇, using

(u,Â u+v)-construction [i] based on
1. digital (4, 37, 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 (5, 72, 68)-net over F₂₇, using
  - net from sequence [i] based on digital (5, 67)-sequence over F₂₇, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₇ with g(F) = 5 and N(F) ≥ 68, using
      - a function field by SÃ©mirat [i]

(42, 109, 224)-Net in Base 27 — Constructive

(42, 109, 224)-net in base 27, using

27¹ times duplication [i] based on (41, 108, 224)-net in base 27, using
- t-expansion [i] based on (40, 108, 224)-net in base 27, using
  - base change [i] based on digital (13, 81, 224)-net over F₈₁, using
    - net from sequence [i] based on digital (13, 223)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 13 and N(F) ≥ 224, using
        a function field by SÃ©mirat [i]

(42, 109, 280)-Net over F₂₇ — Digital

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]

(42, 109, 298)-Net in Base 27

(42, 109, 298)-net in base 27, using

27¹ times duplication [i] based on (41, 108, 298)-net in base 27, using
- t-expansion [i] based on (39, 108, 298)-net in base 27, using
  - base change [i] based on digital (12, 81, 298)-net over F₈₁, using
    - net from sequence [i] based on digital (12, 297)-sequence over F₈₁, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈₁ with g(F) = 12 and N(F) ≥ 298, using
        a curve from the manYPoints database [i]

(42, 109, 24464)-Net in Base 27 — Upper bound on s

There is no (42, 109, 24465)-net in base 27, because

1 times m-reduction [i] would yield (42, 108, 24465)-net in base 27, but
- the generalized Rao bound for nets shows that 27^mÂ â‰¥ 38684 617157 329616 600621 983368 693652 531690 820075 598661 740881 144457 864746 782580 825424 884404 192402 751528 017918 718640 120254 413524 629459 519678 955664 561301 276475 > 27¹⁰⁸ [i]