MinT - Best Known (138−34, 138, s)-Nets in Base 8

Best Known (138−34, 138, s)-Nets in Base 8

(138−34, 138, 1026)-Net over F₈ — Constructive and digital

Digital (104, 138, 1026)-net over F₈, using

14 times m-reduction [i] based on digital (104, 152, 1026)-net over F₈, using
- trace code for nets [i] based on digital (28, 76, 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]

(138−34, 138, 1028)-Net in Base 8 — Constructive

(104, 138, 1028)-net in base 8, using

8² times duplication [i] based on (102, 136, 1028)-net in base 8, using
- base change [i] based on digital (68, 102, 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 (34, 68, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 34, 257)-net over F₂₅₆, using
        net from sequence [i] based on digital (0, 256)-sequence over F₂₅₆ (see above)

(138−34, 138, 11258)-Net over F₈ — Digital

Digital (104, 138, 11258)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(138−34, 138, large)-Net in Base 8 — Upper bound on s

There is no (104, 138, large)-net in base 8, because

32 times m-reduction [i] would yield (104, 106, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]