MinT - Best Known (89, 144, s)-Nets in Base 8

Best Known (89, 144, s)-Nets in Base 8

(89, 144, 354)-Net over F₈ — Constructive and digital

Digital (89, 144, 354)-net over F₈, using

20 times m-reduction [i] based on digital (89, 164, 354)-net over F₈, using
- trace code for nets [i] based on digital (7, 82, 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]

(89, 144, 432)-Net in Base 8 — Constructive

(89, 144, 432)-net in base 8, using

2 times m-reduction [i] based on (89, 146, 432)-net in base 8, using
- trace code for nets [i] based on (16, 73, 216)-net in base 64, using
  - 4 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]

(89, 144, 775)-Net over F₈ — Digital

Digital (89, 144, 775)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(89, 144, 94681)-Net in Base 8 — Upper bound on s

There is no (89, 144, 94682)-net in base 8, because

1 times m-reduction [i] would yield (89, 143, 94682)-net in base 8, but
- the generalized Rao bound for nets shows that 8^mÂ â‰¥ 1386 666731 207947 004248 759213 664352 803171 164930 390886 145666 385054 901348 854839 279811 173707 704440 064772 371082 481815 082532 622993 032672 > 8¹⁴³ [i]