MinT - Best Known (111−31, 111, s)-Nets in Base 16

Best Known (111−31, 111, s)-Nets in Base 16

(111−31, 111, 1285)-Net over F₁₆ — Constructive and digital

Digital (80, 111, 1285)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (9, 19, 257)-net over F₁₆, using
  - base reduction for projective spaces (embedding PG(9,256) in PG(18,16)) 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)

(111−31, 111, 22921)-Net over F₁₆ — Digital

Digital (80, 111, 22921)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(111−31, 111, large)-Net in Base 16 — Upper bound on s

There is no (80, 111, large)-net in base 16, because

29 times m-reduction [i] would yield (80, 82, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]