MinT - Best Known (17, 39, s)-Nets in Base 16

Best Known (17, 39, s)-Nets in Base 16

(17, 39, 82)-Net over F₁₆ — Constructive and digital

Digital (17, 39, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 11, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 28, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]

(17, 39, 112)-Net over F₁₆ — Digital

Digital (17, 39, 112)-net over F₁₆, using

net from sequence [i] based on digital (17, 111)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 17 and N(F) ≥ 112, using
  - a curve from the manYPoints database [i]

(17, 39, 129)-Net in Base 16 — Constructive

(17, 39, 129)-net in base 16, using

base change [i] based on (4, 26, 129)-net in base 64, using
- 2 times m-reduction [i] based on (4, 28, 129)-net in base 64, using
  - base change [i] based on digital (0, 24, 129)-net over F₁₂₈, using
    - net from sequence [i] based on digital (0, 128)-sequence over F₁₂₈, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 0 and N(F) ≥ 129, using
        the rational function field F₁₂₈(x) [i]
      - Niederreiter sequence [i]

(17, 39, 6076)-Net in Base 16 — Upper bound on s

There is no (17, 39, 6077)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 91397 335106 053893 322397 513747 175561 898190 582956 > 16³⁹ [i]