MinT - Best Known (53, 53+32, s)-Nets in Base 16

Best Known (53, 53+32, s)-Nets in Base 16

(53, 53+32, 563)-Net over F₁₆ — Constructive and digital

Digital (53, 85, 563)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (5, 21, 49)-net over F₁₆, using
  - net from sequence [i] based on digital (5, 48)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 5 and N(F) ≥ 49, using
      - a function field by SÃ©mirat [i]
2. digital (32, 64, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 32, 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]

(53, 53+32, 1673)-Net over F₁₆ — Digital

Digital (53, 85, 1673)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(53, 53+32, 1130658)-Net in Base 16 — Upper bound on s

There is no (53, 85, 1130659)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 239749 786963 558723 228364 947422 731352 411241 922131 277768 569592 356120 572907 852774 774550 846365 185220 126286 > 16⁸⁵ [i]