MinT - Best Known (123−36, 123, s)-Nets in Base 16

Best Known (123−36, 123, s)-Nets in Base 16

(123−36, 123, 1066)-Net over F₁₆ — Constructive and digital

Digital (87, 123, 1066)-net over F₁₆, using

generalized (u,Â u+v)-construction [i] based on
1. digital (3, 15, 38)-net over F₁₆, using
  - net from sequence [i] based on digital (3, 37)-sequence over F₁₆, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 3 and N(F) ≥ 38, using
      - a function field by SÃ©mirat [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 (36, 72, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 36, 257)-net over F₂₅₆, using
    - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(123−36, 123, 15822)-Net over F₁₆ — Digital

Digital (87, 123, 15822)-net over F₁₆, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−36, 123, large)-Net in Base 16 — Upper bound on s

There is no (87, 123, large)-net in base 16, because

34 times m-reduction [i] would yield (87, 89, large)-net in base 16, but
- mutually orthogonal hypercube bound [i]