MinT - Best Known (48−26, 48, s)-Nets in Base 16

Best Known (48−26, 48, s)-Nets in Base 16

(48−26, 48, 103)-Net over F₁₆ — Constructive and digital

Digital (22, 48, 103)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 16, 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, 32, 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]

(48−26, 48, 140)-Net over F₁₆ — Digital

Digital (22, 48, 140)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(48−26, 48, 150)-Net in Base 16 — Constructive

(22, 48, 150)-net in base 16, using

1 times m-reduction [i] based on (22, 49, 150)-net in base 16, using
- base change [i] based on digital (1, 28, 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]

(48−26, 48, 161)-Net in Base 16

(22, 48, 161)-net in base 16, using

base change [i] based on digital (6, 32, 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]

(48−26, 48, 10544)-Net in Base 16 — Upper bound on s

There is no (22, 48, 10545)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6283 277606 214941 123124 172021 038492 931742 963786 970898 961776 > 16⁴⁸ [i]