MinT - Best Known (52, 52+27, s)-Nets in Base 8

Best Known (52, 52+27, s)-Nets in Base 8

(52, 52+27, 354)-Net over F₈ — Constructive and digital

Digital (52, 79, 354)-net over F₈, using

11 times m-reduction [i] based on digital (52, 90, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 45, 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+27, 518)-Net in Base 8 — Constructive

(52, 79, 518)-net in base 8, using

1 times m-reduction [i] based on (52, 80, 518)-net in base 8, using
- base change [i] based on digital (32, 60, 518)-net over F₁₆, using
  - trace code for nets [i] based on digital (2, 30, 259)-net over F₂₅₆, using
    - net from sequence [i] based on digital (2, 258)-sequence over F₂₅₆, using
      - Niederreiter sequence [i]

(52, 52+27, 849)-Net over F₈ — Digital

Digital (52, 79, 849)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(52, 52+27, 212242)-Net in Base 8 — Upper bound on s

There is no (52, 79, 212243)-net in base 8, because

1 times m-reduction [i] would yield (52, 78, 212243)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 27608 136091 344536 796842 443669 319465 844830 335833 371257 772147 678453 321832 > 8⁷⁸ [i]