MinT - Best Known (78, 78+24, s)-Nets in Base 8

Best Known (78, 78+24, s)-Nets in Base 8

(78, 78+24, 708)-Net over F₈ — Constructive and digital

Digital (78, 102, 708)-net over F₈, using

t-expansion [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+24, 1032)-Net in Base 8 — Constructive

(78, 102, 1032)-net in base 8, using

trace code for nets [i] based on (27, 51, 516)-net in base 64, using
- 1 times m-reduction [i] based on (27, 52, 516)-net in base 64, using
  - base change [i] based on digital (14, 39, 516)-net over F₂₅₆, using
    - (u,Â u+v)-construction [i] based on
      1. digital (1, 13, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]
      2. digital (1, 26, 258)-net over F₂₅₆, using
        net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆ (see above)

(78, 78+24, 13637)-Net over F₈ — Digital

Digital (78, 102, 13637)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(78, 78+24, large)-Net in Base 8 — Upper bound on s

There is no (78, 102, large)-net in base 8, because

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