MinT - Best Known (55−30, 55, s)-Nets in Base 16

Best Known (55−30, 55, s)-Nets in Base 16

(55−30, 55, 110)-Net over F₁₆ — Constructive and digital

Digital (25, 55, 110)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 19, 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, 36, 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]

(55−30, 55, 147)-Net over F₁₆ — Digital

Digital (25, 55, 147)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(55−30, 55, 150)-Net in Base 16 — Constructive

(25, 55, 150)-net in base 16, using

1 times m-reduction [i] based on (25, 56, 150)-net in base 16, using
- base change [i] based on digital (1, 32, 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]

(55−30, 55, 161)-Net in Base 16

(25, 55, 161)-net in base 16, using

2 times m-reduction [i] based on (25, 57, 161)-net in base 16, using
- base change [i] based on digital (6, 38, 161)-net over F₆₄, using
  - net from sequence [i] based on digital (6, 160)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 6 and N(F) ≥ 161, using
      - a curve from the manYPoints database [i]

(55−30, 55, 11129)-Net in Base 16 — Upper bound on s

There is no (25, 55, 11130)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 1 686462 139935 139432 092516 917714 713056 941718 953404 118311 485719 352376 > 16⁵⁵ [i]