MinT - Best Known (52−20, 52, s)-Nets in Base 16

Best Known (52−20, 52, s)-Nets in Base 16

(52−20, 52, 547)-Net over F₁₆ — Constructive and digital

Digital (32, 52, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 12, 33)-net over F₁₆, using
  - net from sequence [i] based on digital (2, 32)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 2 and N(F) ≥ 33, using
      - a function field by SÃ©mirat [i]
2. digital (20, 40, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 20, 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]

(52−20, 52, 1053)-Net over F₁₆ — Digital

Digital (32, 52, 1053)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(52−20, 52, 551194)-Net in Base 16 — Upper bound on s

There is no (32, 52, 551195)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 411 379797 203691 867523 954570 084792 097829 465221 385547 279847 661751 > 16⁵² [i]