MinT - Best Known (71, 71+45, s)-Nets in Base 16

Best Known (71, 71+45, s)-Nets in Base 16

(71, 71+45, 559)-Net over F₁₆ — Constructive and digital

Digital (71, 116, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 26, 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 (45, 90, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 45, 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]

(71, 71+45, 1741)-Net over F₁₆ — Digital

Digital (71, 116, 1741)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(71, 71+45, 1188559)-Net in Base 16 — Upper bound on s

There is no (71, 116, 1188560)-net in base 16, because

1 times m-reduction [i] would yield (71, 115, 1188560)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 977156 948277 316077 847726 770110 831858 285443 577136 003780 729489 954253 053156 216781 671167 731704 442768 875613 365980 313382 606495 676672 541302 179676 > 16¹¹⁵ [i]