MinT - Best Known (66, 66+42, s)-Nets in Base 16

Best Known (66, 66+42, s)-Nets in Base 16

(66, 66+42, 552)-Net over F₁₆ — Constructive and digital

Digital (66, 108, 552)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (3, 24, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [i]
2. digital (42, 84, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 42, 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, 66+42, 1619)-Net over F₁₆ — Digital

Digital (66, 108, 1619)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(66, 66+42, 901597)-Net in Base 16 — Upper bound on s

There is no (66, 108, 901598)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11090 715827 547902 582046 052335 499572 530304 732668 868671 649669 976244 021683 801852 492531 254171 725466 067920 799509 434912 987958 509423 369371 > 16¹⁰⁸ [i]