MinT - Best Known (128−32, 128, s)-Nets in Base 8

Best Known (128−32, 128, s)-Nets in Base 8

(128−32, 128, 1026)-Net over F₈ — Constructive and digital

Digital (96, 128, 1026)-net over F₈, using

8 times m-reduction [i] based on digital (96, 136, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 68, 513)-net over F₆₄, using
  - net from sequence [i] based on digital (28, 512)-sequence over F₆₄, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 28 and N(F) ≥ 513, using
      - the Hermitian function field over F₆₄ [i]

(128−32, 128, 1028)-Net in Base 8 — Constructive

(96, 128, 1028)-net in base 8, using

base change [i] based on digital (64, 96, 1028)-net over F₁₆, using
- (u,Â u+v)-construction [i] based on
  1. digital (16, 32, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 16, 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 (32, 64, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 32, 257)-net over F₂₅₆, using
      - net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(128−32, 128, 9518)-Net over F₈ — Digital

Digital (96, 128, 9518)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(128−32, 128, large)-Net in Base 8 — Upper bound on s

There is no (96, 128, large)-net in base 8, because

30 times m-reduction [i] would yield (96, 98, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]