MinT - Best Known (27, 27+37, s)-Nets in Base 16

Best Known (27, 27+37, s)-Nets in Base 16

(27, 27+37, 103)-Net over F₁₆ — Constructive and digital

Digital (27, 64, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 21, 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]
2. digital (6, 43, 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+37, 128)-Net in Base 16 — Constructive

(27, 64, 128)-net in base 16, using

2 times m-reduction [i] based on (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+37, 156)-Net over F₁₆ — Digital

Digital (27, 64, 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+37, 8240)-Net in Base 16 — Upper bound on s

There is no (27, 64, 8241)-net in base 16, because

1 times m-reduction [i] would yield (27, 63, 8241)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 7246 581494 869916 925724 135208 094546 136661 989277 807607 644679 158879 453646 756496 > 16⁶³ [i]