MinT - Best Known (110−41, 110, s)-Nets in Base 8

Best Known (110−41, 110, s)-Nets in Base 8

(110−41, 110, 354)-Net over F₈ — Constructive and digital

Digital (69, 110, 354)-net over F₈, using

14 times m-reduction [i] based on digital (69, 124, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 62, 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]

(110−41, 110, 514)-Net in Base 8 — Constructive

(69, 110, 514)-net in base 8, using

trace code for nets [i] based on (14, 55, 257)-net in base 64, using
- 1 times m-reduction [i] based on (14, 56, 257)-net in base 64, using
  - base change [i] based on digital (0, 42, 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]

(110−41, 110, 706)-Net over F₈ — Digital

Digital (69, 110, 706)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(110−41, 110, 99081)-Net in Base 8 — Upper bound on s

There is no (69, 110, 99082)-net in base 8, because

1 times m-reduction [i] would yield (69, 109, 99082)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 273 410150 087005 469054 253101 018067 006323 107057 725257 398258 395086 595490 243726 633119 474091 192073 076832 > 8¹⁰⁹ [i]