MinT - Best Known (108−40, 108, s)-Nets in Base 16

Best Known (108−40, 108, s)-Nets in Base 16

(108−40, 108, 581)-Net over F₁₆ — Constructive and digital

Digital (68, 108, 581)-net over F₁₆, using

(u,Â u+v)-construction [i] based on
1. digital (6, 26, 65)-net over F₁₆, using
  - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
      - the Hermitian function field over F₁₆ [i]
2. digital (42, 82, 516)-net over F₁₆, using
  - trace code for nets [i] based on digital (1, 41, 258)-net over F₂₅₆, using
    - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(108−40, 108, 2237)-Net over F₁₆ — Digital

Digital (68, 108, 2237)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(108−40, 108, 1759787)-Net in Base 16 — Upper bound on s

There is no (68, 108, 1759788)-net in base 16, because

the generalized Rao bound for nets shows that 16^mÂ â‰¥ 11090 754015 734742 861161 345040 380571 069534 582556 660416 006730 901185 478657 669974 408768 743716 387177 466401 958126 714808 473759 387840 667276 > 16¹⁰⁸ [i]