MinT - Best Known (27, 27+45, s)-Nets in Base 16

Best Known (27, 27+45, s)-Nets in Base 16

(27, 27+45, 71)-Net over F₁₆ — Constructive and digital

Digital (27, 72, 71)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 24, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (3, 48, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]

(27, 27+45, 120)-Net in Base 16 — Constructive

(27, 72, 120)-net in base 16, using

8 times m-reduction [i] based on (27, 80, 120)-net in base 16, using
- base change [i] based on digital (11, 64, 120)-net over F₃₂, using
  - net from sequence [i] based on digital (11, 119)-sequence over F₃₂, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₃₂ with g(F) = 11 and N(F) ≥ 120, using
      - a function field by SÃ©mirat [i]

(27, 27+45, 156)-Net over F₁₆ — Digital

Digital (27, 72, 156)-net over F₁₆, using

net from sequence [i] based on digital (27, 155)-sequence over F₁₆, using
- Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 27 and N(F) ≥ 156, using
  - a curve from the manYPoints database [i]

(27, 27+45, 4630)-Net in Base 16 — Upper bound on s

There is no (27, 72, 4631)-net in base 16, because

1 times m-reduction [i] would yield (27, 71, 4631)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 31 097635 192994 899469 061202 640589 403650 768151 065964 139350 253351 842364 312573 980862 614081 > 16⁷¹ [i]