MinT - Best Known (65, 65+43, s)-Nets in Base 16

Best Known (65, 65+43, s)-Nets in Base 16

(65, 65+43, 538)-Net over F₁₆ — Constructive and digital

Digital (65, 108, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 22, 24)-net over F₁₆, using
  - net from sequence [i] based on digital (1, 23)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 1 and N(F) ≥ 24, using
      - a function field by SÃ©mirat [i]
2. digital (43, 86, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 43, 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+43, 1395)-Net over F₁₆ — Digital

Digital (65, 108, 1395)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(65, 65+43, 790083)-Net in Base 16 — Upper bound on s

There is no (65, 108, 790084)-net in base 16, because

1 times m-reduction [i] would yield (65, 107, 790084)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 693 174738 883854 263560 188122 249674 931901 738346 747239 750258 913853 943163 499074 360749 228754 831134 991060 351363 973640 757127 616373 168336 > 16¹⁰⁷ [i]