MinT - Best Known (128−37, 128, s)-Nets in Base 16

Best Known (128−37, 128, s)-Nets in Base 16

(128−37, 128, 1093)-Net over F₁₆ — Constructive and digital

Digital (91, 128, 1093)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (6, 18, 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 (18, 36, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 18, 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]
3. digital (37, 74, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 37, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(128−37, 128, 18213)-Net over F₁₆ — Digital

Digital (91, 128, 18213)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−37, 128, large)-Net in Base 16 — Upper bound on s

There is no (91, 128, large)-net in base 16, because

35 times m-reduction [i] would yield (91, 93, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]