MinT - Best Known (144−52, 144, s)-Nets in Base 8

Best Known (144−52, 144, s)-Nets in Base 8

(144−52, 144, 363)-Net over F₈ — Constructive and digital

Digital (92, 144, 363)-net over F₈, using

(u,Â u+v)-construction [i] based on
1. digital (0, 26, 9)-net over F₈, using
  - net from sequence [i] based on digital (0, 8)-sequence over F₈, using
    - generalized Faure sequence [i]
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₈ with g(F) = 0 and N(F) ≥ 9, using
      - the rational function field F₈(x) [i]
    - Niederreiter sequence [i]
2. digital (66, 118, 354)-net over F₈, using
  - trace code for nets [i] based on digital (7, 59, 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]

(144−52, 144, 576)-Net in Base 8 — Constructive

(92, 144, 576)-net in base 8, using

trace code for nets [i] based on (20, 72, 288)-net in base 64, using
- 5 times m-reduction [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]

(144−52, 144, 1032)-Net over F₈ — Digital

Digital (92, 144, 1032)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(144−52, 144, 151313)-Net in Base 8 — Upper bound on s

There is no (92, 144, 151314)-net in base 8, because

the generalized Rao bound for nets shows that 8^mÂ â‰¥ 11091 666339 708431 230926 758776 628126 235422 126361 133847 763641 562524 516988 741140 118440 725628 101358 696229 466973 122380 413461 462868 213888 > 8¹⁴⁴ [i]