MinT - Best Known (47−27, 47, s)-Nets in Base 16

Best Known (47−27, 47, s)-Nets in Base 16

(47−27, 47, 89)-Net over F₁₆ — Constructive and digital

Digital (20, 47, 89)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 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, 33, 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]

(47−27, 47, 104)-Net in Base 16 — Constructive

(20, 47, 104)-net in base 16, using

8 times m-reduction [i] based on (20, 55, 104)-net in base 16, using
- base change [i] based on digital (9, 44, 104)-net over F₃₂, using
  - net from sequence [i] based on digital (9, 103)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 9 and N(F) ≥ 104, using
      - a function field by SÃ©mirat [i]

(47−27, 47, 129)-Net over F₁₆ — Digital

Digital (20, 47, 129)-net over F₁₆, using

t-expansion [i] based on digital (19, 47, 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]

(47−27, 47, 6880)-Net in Base 16 — Upper bound on s

There is no (20, 47, 6881)-net in base 16, because

1 times m-reduction [i] would yield (20, 46, 6881)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 24 547432 199854 419891 300721 877131 182316 993140 320081 608096 > 16⁴⁶ [i]