MinT - Best Known (82, 82+31, s)-Nets in Base 16

Best Known (82, 82+31, s)-Nets in Base 16

(82, 82+31, 1542)-Net over F₁₆ — Constructive and digital

Digital (82, 113, 1542)-net over F₁₆, using

16¹ times duplication [i] based on digital (81, 112, 1542)-net over F₁₆, using
- generalized (u,Â u+v)-construction [i] based on
  1. digital (10, 20, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 10, 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]
  2. digital (15, 30, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 15, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)
  3. digital (31, 62, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 31, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(82, 82+31, 27571)-Net over F₁₆ — Digital

Digital (82, 113, 27571)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(82, 82+31, large)-Net in Base 16 — Upper bound on s

There is no (82, 113, large)-net in base 16, because

29 times m-reduction [i] would yield (82, 84, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]