MinT - Best Known (74−23, 74, s)-Nets in Base 8

Best Known (74−23, 74, s)-Nets in Base 8

(74−23, 74, 378)-Net over F₈ — Constructive and digital

Digital (51, 74, 378)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (3, 14, 24)-net over F₈, using
  - net from sequence [i] based on digital (3, 23)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 3 and N(F) ≥ 24, using
      - the Klein quartic over F₈ [i]
2. digital (37, 60, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 30, 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]

(74−23, 74, 528)-Net in Base 8 — Constructive

(51, 74, 528)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 12, 14)-net over F₈, using
  - net from sequence [i] based on digital (1, 13)-sequence over F₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 1 and N(F) ≥ 14, using
      - F₂ from the tower of function fields over F₈ by van der Geer and van der Vlugt [i]
2. (39, 62, 514)-net in base 8, using
  - trace code for nets [i] based on (8, 31, 257)-net in base 64, using
    - 1 times m-reduction [i] based on (8, 32, 257)-net in base 64, using
      - base change [i] based on digital (0, 24, 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]

(74−23, 74, 1422)-Net over F₈ — Digital

Digital (51, 74, 1422)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(74−23, 74, 690471)-Net in Base 8 — Upper bound on s

There is no (51, 74, 690472)-net in base 8, because

1 times m-reduction [i] would yield (51, 73, 690472)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 842509 831923 590891 737527 004860 818020 899131 666530 455693 287366 977460 > 8⁷³ [i]