MinT - Best Known (18, 42, s)-Nets in Base 16

Best Known (18, 42, s)-Nets in Base 16

(18, 42, 82)-Net over F₁₆ — Constructive and digital

Digital (18, 42, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 12, 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, 30, 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]

(18, 42, 113)-Net over F₁₆ — Digital

Digital (18, 42, 113)-net over F₁₆, using

net from sequence [i] based on digital (18, 112)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 18 and N(F) ≥ 113, using
  - a curve from the manYPoints database [i]

(18, 42, 129)-Net in Base 16 — Constructive

(18, 42, 129)-net in base 16, 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]

(18, 42, 5770)-Net in Base 16 — Upper bound on s

There is no (18, 42, 5771)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 374 503189 419742 122808 360845 466682 663009 036887 452256 > 16⁴² [i]