MinT - Best Known (142−45, 142, s)-Nets in Base 8

Best Known (142−45, 142, s)-Nets in Base 8

(142−45, 142, 419)-Net over F₈ — Constructive and digital

Digital (97, 142, 419)-net over F₈, using

1 times m-reduction [i] based on digital (97, 143, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 37, 65)-net over F₈, using
    - net from sequence [i] based on digital (14, 64)-sequence over F₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 14 and N(F) ≥ 65, using
        the Suzuki function field over F₈ [i]
  2. digital (60, 106, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 53, 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]

(142−45, 142, 576)-Net in Base 8 — Constructive

(97, 142, 576)-net in base 8, using

12 times m-reduction [i] based on (97, 154, 576)-net in base 8, using
- trace code for nets [i] based on (20, 77, 288)-net in base 64, using
  - base change [i] based on digital (9, 66, 288)-net over F₁₂₈, using
    - net from sequence [i] based on digital (9, 287)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 9 and N(F) ≥ 288, using
        a function field by SÃ©mirat [i]

(142−45, 142, 2047)-Net over F₈ — Digital

Digital (97, 142, 2047)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(142−45, 142, 793840)-Net in Base 8 — Upper bound on s

There is no (97, 142, 793841)-net in base 8, because

1 times m-reduction [i] would yield (97, 141, 793841)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 21 661675 573945 255104 354451 461680 274928 202575 316830 959967 557154 321648 837566 876279 896930 688091 428635 004932 620199 557091 023716 744912 > 8¹⁴¹ [i]