MinT - Best Known (66−26, 66, s)-Nets in Base 16

Best Known (66−26, 66, s)-Nets in Base 16

(66−26, 66, 538)-Net over F₁₆ — Constructive and digital

Digital (40, 66, 538)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (1, 14, 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 (26, 52, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 26, 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]

(66−26, 66, 1037)-Net over F₁₆ — Digital

Digital (40, 66, 1037)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(66−26, 66, 490380)-Net in Base 16 — Upper bound on s

There is no (40, 66, 490381)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 29 643014 163086 720536 893931 794318 406541 002967 086822 419864 188662 494417 261968 278096 > 16⁶⁶ [i]