MinT - Best Known (65, 65+40, s)-Nets in Base 16

Best Known (65, 65+40, s)-Nets in Base 16

(65, 65+40, 563)-Net over F₁₆ — Constructive and digital

Digital (65, 105, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 25, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (40, 80, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 40, 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+40, 1811)-Net over F₁₆ — Digital

Digital (65, 105, 1811)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+40, 1161023)-Net in Base 16 — Upper bound on s

There is no (65, 105, 1161024)-net in base 16, because

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]