MinT - Best Known (52, 52+25, s)-Nets in Base 8

Best Known (52, 52+25, s)-Nets in Base 8

(52, 52+25, 368)-Net over F₈ — Constructive and digital

Digital (52, 77, 368)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (1, 13, 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. 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, 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]

(52, 52+25, 520)-Net in Base 8 — Constructive

(52, 77, 520)-net in base 8, using

1 times m-reduction [i] based on (52, 78, 520)-net in base 8, using
- trace code for nets [i] based on (13, 39, 260)-net in base 64, using
  - 1 times m-reduction [i] based on (13, 40, 260)-net in base 64, using
    - base change [i] based on digital (3, 30, 260)-net over F₂₅₆, using
      - net from sequence [i] based on digital (3, 259)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(52, 52+25, 1118)-Net over F₈ — Digital

Digital (52, 77, 1118)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+25, 396118)-Net in Base 8 — Upper bound on s

There is no (52, 77, 396119)-net in base 8, because

1 times m-reduction [i] would yield (52, 76, 396119)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 431 362628 982841 772943 188438 599422 234562 634117 746147 061412 388710 195694 > 8⁷⁶ [i]