MinT - Best Known (65, 65+41, s)-Nets in Base 16

Best Known (65, 65+41, s)-Nets in Base 16

(65, 65+41, 559)-Net over F₁₆ — Constructive and digital

Digital (65, 106, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 24, 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 (41, 82, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 41, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆, using
      - generalized Faure sequence [i]
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₂₅₆ with g(F) = 0 and N(F) ≥ 257, using
        the rational function field F₂₅₆(x) [i]
      - Niederreiter sequence [i]

(65, 65+41, 1652)-Net over F₁₆ — Digital

Digital (65, 106, 1652)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+41, 1161023)-Net in Base 16 — Upper bound on s

There is no (65, 106, 1161024)-net in base 16, because

1 times m-reduction [i] would yield (65, 105, 1161024)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 707731 245866 008657 874684 444020 956706 571253 565576 306219 262766 490061 546724 479085 018529 416850 751883 100335 726729 145781 227145 319701 > 16¹⁰⁵ [i]