MinT - Best Known (123−37, 123, s)-Nets in Base 8

Best Known (123−37, 123, s)-Nets in Base 8

(123−37, 123, 419)-Net over F₈ — Constructive and digital

Digital (86, 123, 419)-net over F₈, using

2 times m-reduction [i] based on digital (86, 125, 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 (53, 92, 354)-net over F₈, using
    - trace code for nets [i] based on digital (7, 46, 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]

(123−37, 123, 576)-Net in Base 8 — Constructive

(86, 123, 576)-net in base 8, using

11 times m-reduction [i] based on (86, 134, 576)-net in base 8, using
- trace code for nets [i] based on (19, 67, 288)-net in base 64, using
  - 3 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]

(123−37, 123, 2502)-Net over F₈ — Digital

Digital (86, 123, 2502)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(123−37, 123, 1425515)-Net in Base 8 — Upper bound on s

There is no (86, 123, 1425516)-net in base 8, because

1 times m-reduction [i] would yield (86, 122, 1425516)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 150 308016 950239 192312 886952 373376 018758 011509 459944 381452 303073 203315 901363 839124 879765 070064 487826 969855 669050 > 8¹²² [i]