MinT - Best Known (48−15, 48, s)-Nets in Base 8

Best Known (48−15, 48, s)-Nets in Base 8

(48−15, 48, 354)-Net over F₈ — Constructive and digital

Digital (33, 48, 354)-net over F₈, using

4 times m-reduction [i] based on digital (33, 52, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 26, 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]

(48−15, 48, 528)-Net in Base 8 — Constructive

(33, 48, 528)-net in base 8, using

(u,Â u+v)-construction [i] based on
1. digital (1, 8, 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. (25, 40, 514)-net in base 8, using
  - base change [i] based on digital (15, 30, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 15, 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]

(48−15, 48, 1085)-Net over F₈ — Digital

Digital (33, 48, 1085)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(48−15, 48, 559012)-Net in Base 8 — Upper bound on s

There is no (33, 48, 559013)-net in base 8, because

1 times m-reduction [i] would yield (33, 47, 559013)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 2 787602 530402 868419 327524 270805 657337 075424 > 8⁴⁷ [i]