MinT - Best Known (77, 77+45, s)-Nets in Base 8

Best Known (77, 77+45, s)-Nets in Base 8

(77, 77+45, 354)-Net over F₈ — Constructive and digital

Digital (77, 122, 354)-net over F₈, using

18 times m-reduction [i] based on digital (77, 140, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 70, 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]

(77, 77+45, 514)-Net in Base 8 — Constructive

(77, 122, 514)-net in base 8, using

8² times duplication [i] based on (75, 120, 514)-net in base 8, using
- base change [i] based on digital (45, 90, 514)-net over F₁₆, using
  - trace code for nets [i] based on digital (0, 45, 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]

(77, 77+45, 809)-Net over F₈ — Digital

Digital (77, 122, 809)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(77, 77+45, 119867)-Net in Base 8 — Upper bound on s

There is no (77, 122, 119868)-net in base 8, because

1 times m-reduction [i] would yield (77, 121, 119868)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 18 790976 042565 450675 219040 769208 109124 897439 273331 989472 714275 572072 870065 412699 691380 915363 770682 436323 791566 > 8¹²¹ [i]