MinT - Best Known (77−45, 77, s)-Nets in Base 16

Best Known (77−45, 77, s)-Nets in Base 16

(77−45, 77, 110)-Net over F₁₆ — Constructive and digital

Digital (32, 77, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 26, 45)-net over F₁₆, using
  - net from sequence [i] based on digital (4, 44)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 4 and N(F) ≥ 45, using
      - a function field by GarcÃa and Quoos [i]
2. digital (6, 51, 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]

(77−45, 77, 128)-Net in Base 16 — Constructive

(32, 77, 128)-net in base 16, using

4 times m-reduction [i] based on (32, 81, 128)-net in base 16, using
- base change [i] based on digital (5, 54, 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]

(77−45, 77, 168)-Net over F₁₆ — Digital

Digital (32, 77, 168)-net over F₁₆, using

t-expansion [i] based on digital (31, 77, 168)-net over F₁₆, using
- net from sequence [i] based on digital (31, 167)-sequence over F₁₆, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 31 and N(F) ≥ 168, using
    - a curve from the manYPoints database [i]

(77−45, 77, 8706)-Net in Base 16 — Upper bound on s

There is no (32, 77, 8707)-net in base 16, because

1 times m-reduction [i] would yield (32, 76, 8707)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 32 618718 571329 639726 899709 455270 759076 300640 232939 719900 635089 993543 872248 333767 163502 596736 > 16⁷⁶ [i]