MinT - Best Known (51−20, 51, s)-Nets in Base 16

Best Known (51−20, 51, s)-Nets in Base 16

(51−20, 51, 538)-Net over F₁₆ — Constructive and digital

Digital (31, 51, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 11, 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 (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]

(51−20, 51, 912)-Net over F₁₆ — Digital

Digital (31, 51, 912)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(51−20, 51, 417726)-Net in Base 16 — Upper bound on s

There is no (31, 51, 417727)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 25 711578 693498 355849 295332 397937 024240 334256 526442 548173 500051 > 16⁵¹ [i]