MinT - Best Known (117−42, 117, s)-Nets in Base 16

Best Known (117−42, 117, s)-Nets in Base 16

(117−42, 117, 585)-Net over F₁₆ — Constructive and digital

Digital (75, 117, 585)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 27, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (48, 90, 520)-net over F₁₆, using
  - trace code for nets [i] based on digital (3, 45, 260)-net over F₂₅₆, using
    - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(117−42, 117, 594)-Net in Base 16 — Constructive

(75, 117, 594)-net in base 16, using

(u,Â u+v)-construction [i] based on
1. (12, 33, 80)-net in base 16, using
  - base change [i] based on digital (1, 22, 80)-net over F₆₄, using
    - net from sequence [i] based on digital (1, 79)-sequence over F₆₄, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 1 and N(F) ≥ 80, using
        a function field by SÃ©mirat [i]
2. digital (42, 84, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 42, 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]

(117−42, 117, 2958)-Net over F₁₆ — Digital

Digital (75, 117, 2958)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(117−42, 117, 2958476)-Net in Base 16 — Upper bound on s

There is no (75, 117, 2958477)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 762 148878 569930 272121 104961 340616 734194 259295 122932 034139 601331 103117 890668 114700 692592 960263 095153 076525 278251 884627 647602 207780 905496 648256 > 16¹¹⁷ [i]