MinT - Best Known (116−43, 116, s)-Nets in Base 16

Best Known (116−43, 116, s)-Nets in Base 16

(116−43, 116, 581)-Net over F₁₆ — Constructive and digital

Digital (73, 116, 581)-net over F₁₆, using

16¹ times duplication [i] based on digital (72, 115, 581)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (6, 27, 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 (45, 88, 516)-net over F₁₆, using
    - trace code for nets [i] based on digital (1, 44, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(116−43, 116, 2351)-Net over F₁₆ — Digital

Digital (73, 116, 2351)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(116−43, 116, 2271901)-Net in Base 16 — Upper bound on s

There is no (73, 116, 2271902)-net in base 16, because

1 times m-reduction [i] would yield (73, 115, 2271902)-net in base 16, but
- the generalized Rao bound for nets shows that 16^mÂ â‰¥ 2 977143 462743 375092 668973 209075 982388 235316 952888 103129 926655 309656 990403 553795 894475 520191 579704 759261 840159 504581 203376 318466 792994 064631 > 16¹¹⁵ [i]