MinT - Best Known (59−23, 59, s)-Nets in Base 16

Best Known (59−23, 59, s)-Nets in Base 16

(59−23, 59, 547)-Net over F₁₆ — Constructive and digital

Digital (36, 59, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 13, 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 (23, 46, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 23, 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]

(59−23, 59, 1034)-Net over F₁₆ — Digital

Digital (36, 59, 1034)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(59−23, 59, 730997)-Net in Base 16 — Upper bound on s

There is no (36, 59, 730998)-net in base 16, because

1 times m-reduction [i] would yield (36, 58, 730998)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 6901 820753 661939 854230 479186 287342 693489 820690 094901 883276 561950 970796 > 16⁵⁸ [i]