MinT - Best Known (27, 27+39, s)-Nets in Base 16

Best Known (27, 27+39, s)-Nets in Base 16

(27, 27+39, 98)-Net over F₁₆ — Constructive and digital

Digital (27, 66, 98)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 21, 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 (6, 45, 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]

(27, 27+39, 128)-Net in Base 16 — Constructive

(27, 66, 128)-net in base 16, using

base change [i] based on digital (5, 44, 128)-net over F₆₄, using
- net from sequence [i] based on digital (5, 127)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 5 and N(F) ≥ 128, using
    - a function field by SÃ©mirat [i]

(27, 27+39, 156)-Net over F₁₆ — Digital

Digital (27, 66, 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+39, 6947)-Net in Base 16 — Upper bound on s

There is no (27, 66, 6948)-net in base 16, because

1 times m-reduction [i] would yield (27, 65, 6948)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 854193 164390 056966 353784 787751 570147 869199 369156 335570 269277 477689 449432 894906 > 16⁶⁵ [i]