MinT - Best Known (77, 77+49, s)-Nets in Base 16

Best Known (77, 77+49, s)-Nets in Base 16

(77, 77+49, 559)-Net over F₁₆ — Constructive and digital

Digital (77, 126, 559)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (4, 28, 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 (49, 98, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 49, 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]

(77, 77+49, 1833)-Net over F₁₆ — Digital

Digital (77, 126, 1833)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+49, 1221026)-Net in Base 16 — Upper bound on s

There is no (77, 126, 1221027)-net in base 16, because

1 times m-reduction [i] would yield (77, 125, 1221027)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 3 273427 974976 333821 318389 432471 338291 152953 334459 947817 201915 986595 752728 196844 965899 701460 895143 366209 676336 513681 723492 434239 176662 022260 455565 214396 > 16¹²⁵ [i]