MinT - Best Known (111−44, 111, s)-Nets in Base 16

Best Known (111−44, 111, s)-Nets in Base 16

(111−44, 111, 538)-Net over F₁₆ — Constructive and digital

Digital (67, 111, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 23, 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 (44, 88, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 44, 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]

(111−44, 111, 1466)-Net over F₁₆ — Digital

Digital (67, 111, 1466)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−44, 111, 717938)-Net in Base 16 — Upper bound on s

There is no (67, 111, 717939)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 45 428287 644699 323963 960952 703891 333306 012299 932346 103992 246671 639222 132370 484939 409448 302761 507588 597799 536985 086896 105340 975880 643496 > 16¹¹¹ [i]