MinT - Best Known (109−39, 109, s)-Nets in Base 8

Best Known (109−39, 109, s)-Nets in Base 8

(109−39, 109, 354)-Net over F₈ — Constructive and digital

Digital (70, 109, 354)-net over F₈, using

17 times m-reduction [i] based on digital (70, 126, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 63, 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]

(109−39, 109, 516)-Net in Base 8 — Constructive

(70, 109, 516)-net in base 8, using

1 times m-reduction [i] based on (70, 110, 516)-net in base 8, using
- trace code for nets [i] based on (15, 55, 258)-net in base 64, using
  - 1 times m-reduction [i] based on (15, 56, 258)-net in base 64, using
    - base change [i] based on digital (1, 42, 258)-net over F₂₅₆, using
      - net from sequence [i] based on digital (1, 257)-sequence over F₂₅₆, using
        Niederreiter sequence [i]

(109−39, 109, 855)-Net over F₈ — Digital

Digital (70, 109, 855)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(109−39, 109, 153970)-Net in Base 8 — Upper bound on s

There is no (70, 109, 153971)-net in base 8, because

1 times m-reduction [i] would yield (70, 108, 153971)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 34 177128 286442 754568 499348 191734 462259 065000 209361 527340 702433 269883 265201 868566 692125 937952 717736 > 8¹⁰⁸ [i]