MinT - Best Known (43−24, 43, s)-Nets in Base 16

Best Known (43−24, 43, s)-Nets in Base 16

(43−24, 43, 89)-Net over F₁₆ — Constructive and digital

Digital (19, 43, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [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]

(43−24, 43, 129)-Net in Base 16 — Constructive

(19, 43, 129)-net in base 16, using

16¹ times duplication [i] based on (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]

(43−24, 43, 129)-Net over F₁₆ — Digital

Digital (19, 43, 129)-net over F₁₆, using

net from sequence [i] based on digital (19, 128)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 19 and N(F) ≥ 129, using
  - a curve from the manYPoints database [i]

(43−24, 43, 7272)-Net in Base 16 — Upper bound on s

There is no (19, 43, 7273)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 5995 598135 175046 201727 817067 132600 462244 913508 554616 > 16⁴³ [i]