MinT - Best Known (144−56, 144, s)-Nets in Base 8

Best Known (144−56, 144, s)-Nets in Base 8

(144−56, 144, 354)-Net over F₈ — Constructive and digital

Digital (88, 144, 354)-net over F₈, using

18 times m-reduction [i] based on digital (88, 162, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 81, 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]

(144−56, 144, 432)-Net in Base 8 — Constructive

(88, 144, 432)-net in base 8, using

trace code for nets [i] based on (16, 72, 216)-net in base 64, using
- 5 times m-reduction [i] based on (16, 77, 216)-net in base 64, using
  - base change [i] based on digital (5, 66, 216)-net over F₁₂₈, using
    - net from sequence [i] based on digital (5, 215)-sequence over F₁₂₈, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 5 and N(F) ≥ 216, using
        a function field by SÃ©mirat [i]

(144−56, 144, 709)-Net over F₈ — Digital

Digital (88, 144, 709)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−56, 144, 71164)-Net in Base 8 — Upper bound on s

There is no (88, 144, 71165)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11093 145307 600373 060964 882965 879406 512084 678438 802042 505922 054971 120341 400080 667883 174957 421102 366743 856935 217840 928394 707002 459855 > 8¹⁴⁴ [i]