MinT - Best Known (67−26, 67, s)-Nets in Base 16

Best Known (67−26, 67, s)-Nets in Base 16

(67−26, 67, 547)-Net over F₁₆ — Constructive and digital

Digital (41, 67, 547)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (2, 15, 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 (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]

(67−26, 67, 1157)-Net over F₁₆ — Digital

Digital (41, 67, 1157)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(67−26, 67, 606958)-Net in Base 16 — Upper bound on s

There is no (41, 67, 606959)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 474 287354 107752 578694 203721 534981 182823 254476 075284 084853 611515 980321 784304 920506 > 16⁶⁷ [i]