MinT - Best Known (78, 78+25, s)-Nets in Base 8

Best Known (78, 78+25, s)-Nets in Base 8

(78, 78+25, 708)-Net over F₈ — Constructive and digital

Digital (78, 103, 708)-net over F₈, using

8¹ times duplication [i] based on digital (77, 102, 708)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (26, 38, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 19, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
        Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
        a function field by GarcÃa and Quoos [i]
  2. digital (39, 64, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 32, 177)-net over F₆₄, using
      - net from sequence [i] based on digital (7, 176)-sequence over F₆₄ (see above)

(78, 78+25, 1030)-Net in Base 8 — Constructive

(78, 103, 1030)-net in base 8, using

8¹ times duplication [i] based on (77, 102, 1030)-net in base 8, using
- (u,Â u+v)-construction [i] based on
  1. (20, 32, 514)-net in base 8, using
    - base change [i] based on digital (12, 24, 514)-net over F₁₆, using
      - trace code for nets [i] based on digital (0, 12, 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. (45, 70, 516)-net in base 8, using
    - trace code for nets [i] based on (10, 35, 258)-net in base 64, using
      - 1 times m-reduction [i] based on (10, 36, 258)-net in base 64, using
        base change [i] based on digital (1, 27, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(78, 78+25, 10532)-Net over F₈ — Digital

Digital (78, 103, 10532)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+25, large)-Net in Base 8 — Upper bound on s

There is no (78, 103, large)-net in base 8, because

23 times m-reduction [i] would yield (78, 80, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]