MinT - Best Known (122−37, 122, s)-Nets in Base 8

Best Known (122−37, 122, s)-Nets in Base 8

(122−37, 122, 419)-Net over F₈ — Constructive and digital

Digital (85, 122, 419)-net over F₈, using

1 times m-reduction [i] based on digital (85, 123, 419)-net over F₈, using
- (u,Â u+v)-construction [i] based on
  1. digital (14, 33, 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 (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]

(122−37, 122, 576)-Net in Base 8 — Constructive

(85, 122, 576)-net in base 8, using

10 times m-reduction [i] based on (85, 132, 576)-net in base 8, using
- trace code for nets [i] based on (19, 66, 288)-net in base 64, using
  - 4 times m-reduction [i] based on (19, 70, 288)-net in base 64, using
    - base change [i] based on digital (9, 60, 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]

(122−37, 122, 2363)-Net over F₈ — Digital

Digital (85, 122, 2363)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(122−37, 122, 1269988)-Net in Base 8 — Upper bound on s

There is no (85, 122, 1269989)-net in base 8, because

1 times m-reduction [i] would yield (85, 121, 1269989)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 18 788457 065205 327049 342218 020549 877946 864337 619960 489013 552879 805099 666648 253850 070063 663385 578188 066971 630140 > 8¹²¹ [i]