MinT - Best Known (140−35, 140, s)-Nets in Base 8

Best Known (140−35, 140, s)-Nets in Base 8

(140−35, 140, 1026)-Net over F₈ — Constructive and digital

Digital (105, 140, 1026)-net over F₈, using

14 times m-reduction [i] based on digital (105, 154, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 77, 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]

(140−35, 140, 1028)-Net in Base 8 — Constructive

(105, 140, 1028)-net in base 8, using

base change [i] based on digital (70, 105, 1028)-net over F₁₆, using
- 16¹ times duplication [i] based on digital (69, 104, 1028)-net over F₁₆, using
  - (u,Â u+v)-construction [i] based on
    1. digital (17, 34, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 17, 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 (35, 70, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 35, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(140−35, 140, 10132)-Net over F₈ — Digital

Digital (105, 140, 10132)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(140−35, 140, large)-Net in Base 8 — Upper bound on s

There is no (105, 140, large)-net in base 8, because

33 times m-reduction [i] would yield (105, 107, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]