MinT - Best Known (28−15, 28, s)-Nets in Base 16

Best Known (28−15, 28, s)-Nets in Base 16

(28−15, 28, 82)-Net over F₁₆ — Constructive and digital

Digital (13, 28, 82)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (0, 7, 17)-net over F₁₆, using
  - net from sequence [i] based on digital (0, 16)-sequence over F₁₆, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 0 and N(F) ≥ 17, using
      - the rational function field F₁₆(x) [i]
    - Niederreiter sequence [i]
2. digital (6, 21, 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]

(28−15, 28, 110)-Net over F₁₆ — Digital

Digital (13, 28, 110)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(28−15, 28, 150)-Net in Base 16 — Constructive

(13, 28, 150)-net in base 16, using

base change [i] based on digital (1, 16, 150)-net over F₁₂₈, using
- net from sequence [i] based on digital (1, 149)-sequence over F₁₂₈, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 1 and N(F) ≥ 150, using
    - a function field by SÃ©mirat [i]

(28−15, 28, 9934)-Net in Base 16 — Upper bound on s

There is no (13, 28, 9935)-net in base 16, because

1 times m-reduction [i] would yield (13, 27, 9935)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 324 671275 783454 858436 054132 971176 > 16²⁷ [i]