MinT - Best Known (123−45, 123, s)-Nets in Base 8

Best Known (123−45, 123, s)-Nets in Base 8

(123−45, 123, 354)-Net over F₈ — Constructive and digital

Digital (78, 123, 354)-net over F₈, using

19 times m-reduction [i] based on digital (78, 142, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 71, 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]

(123−45, 123, 514)-Net in Base 8 — Constructive

(78, 123, 514)-net in base 8, using

1 times m-reduction [i] based on (78, 124, 514)-net in base 8, using
- base change [i] based on digital (47, 93, 514)-net over F₁₆, using
  - 1 times m-reduction [i] based on digital (47, 94, 514)-net over F₁₆, using
    - trace code for nets [i] based on digital (0, 47, 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]

(123−45, 123, 847)-Net over F₈ — Digital

Digital (78, 123, 847)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−45, 123, 131751)-Net in Base 8 — Upper bound on s

There is no (78, 123, 131752)-net in base 8, because

1 times m-reduction [i] would yield (78, 122, 131752)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 327463 791808 768749 302796 473428 076635 664493 961510 616298 081864 821284 893379 636942 579666 929653 416872 876748 826746 > 8¹²² [i]