MinT - Best Known (59, 59+38, s)-Nets in Base 16

Best Known (59, 59+38, s)-Nets in Base 16

(59, 59+38, 547)-Net over F₁₆ — Constructive and digital

Digital (59, 97, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 21, 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 (38, 76, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 38, 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, 59+38, 1420)-Net over F₁₆ — Digital

Digital (59, 97, 1420)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(59, 59+38, 742108)-Net in Base 16 — Upper bound on s

There is no (59, 97, 742109)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 630 434147 527992 294496 484074 916954 060640 742694 876776 879473 133118 019157 508542 085010 014389 252185 035411 607154 869606 264316 > 16⁹⁷ [i]